Popular in Course
Popular in Chemistry
This 56 page Class Notes was uploaded by Aurore Armstrong on Wednesday October 28, 2015. The Class Notes belongs to CHEM 4515 at University of Wyoming taught by Michael Sommer in Fall. Since its upload, it has received 28 views. For similar materials see /class/230373/chem-4515-university-of-wyoming in Chemistry at University of Wyoming.
Reviews for App
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 10/28/15
Butadlene HMO s HMO Analysis ofButadlene7 only the math 175 o o det E 5 0 70 o 175 o o 175 17 9 o 9 0 1017mm 9 175 9 7 deL0 175 9 o 9 175 o 9 175 175 9 9 9 175 9 u7Eu7EdeL u757 dez0 u7 7 zdet 175 7oz 7D ltu751791791lta75gt179 7517191 7lta 7 517975910 7 Egt17lta 7 E1 791 7901 7 D 02 7 E1 791 901 7 5 a7E 9a7E79 70 AND 02751490175 7920 The fourmots are 7 z wepwae w a f97a15189 7 z Q75W J EZ7Q7O39T E97Q05189 E 4 1 3 a75gt53aaf a70518 1 2 1 5757 W Eo ra2lJ h3Q71518 lam assummgm thlsverslonthatthe signs of 11 and are negatwe sommerj Butadiene HMO s Eigen Vector 1 a 13 0 0 a1 a1 fwl Z Z 2 1 a1618 Elz31 0 0 13 0 d1 d1 om1 13121 a 1618Ba1 121161891 3c71c1213c1 0c16183b1 c71c 1161821 gt 31210ccl ol1 0c16183c1 b1d1161861 clowll 0c16183al1 011618011 Substituting for 21 in the second equation and C1 in the third equation 1211618a1 a1 c11618b1gt a1 c116181618a1 2618a1 gt c11618a1 121 4111618c1 121 d1 16181618d1 2618611 1211618d1 c11618d1 Comparing these to the first and fourth equations we see that 121 1618a1 c 1618 b c 1 71 gt 1 1 1211618d1 a1 all c11618d1 which when normalized gives Eigen Vector 2 a 3 0 0 a2 a2 A a B a B 0 b2 172 a H 0618 E 1P2 0 B a B CZ 05 CZ 2102 0 0 B 1 d2 d2 sommer2 Butadiene HMO s 122 0618a2 a2 62 0618172 gt gt We can do the same klnd of substltutlons we dld before 122 d2 0618c2 c2 0618612 a2 c2 06180618a2 0382 c2 0618a2 122 d2 06180618d2 0382612 d2 0618d2 Again comparing with first and fourth lines 122 0618a2 0618 d gt 62 92 gt 2 2 d2 0618d2 122 c2 c2 061842 which gives Notice that this one and the first one are bonding MO s Eigen Vector 3 a 3 0 0 a3 A a a 0 b b a Hng B B 3 oz 061813 3 Em 0 B a B C3 Cs 0 0 B 1 d3 d3 23 0618a3 123 0618a3 0618b 0382 0618 d a3c3 3gta3c3 gtc3 a3zgta3 3 123 d3 0618c3 123 d3 0382d3 123 0618d3 123 c3 c3 0618d3 c3 0618d3 This one and the neXt one are antibonding MO s sommer3 Butadiene HMO S Eigen Vector 4 a 3 0 0 a4 a4 A 3 a 3 0 b4 174 a H a 1618 E 1P4 0 B a B 4 964 W4 0 0 B 0 d4 d4 124 1618a4 124 1618a4 a4 c4 1618b4 a4 c4 2618a4 64 1618a4 a4 d4 gt gt 124 d4 161864 124 d4 268d4 124 1618d4 b4 64 64 1618d4 64 1618d4 sommer4 Linear Algebra Basic idea of Linear Algebra Subject deals with systems of equations that are related linearly It has many applications in multidimensional vector calculus and related fields Definition Systems of Linear Equations A group of equations that share variables The variables can be functions of other variables as long as they are linear simple polynomials example 5x 2y 3z 2 8 2x z 6 22 y 1 All three equations involve the variable x y and z with different coefficients Solving Systems of Linear Equations Determining numerical or functional values of the variables that satisfy ALL of the equations example This is done by finding simple functional relationships between the variables and substituting the functional forms 5x2y 3z8 2xz6 gt x6 z2 22 y1 gt y22 1 Plugging in the two functions of 2 into the first equation 5x 2y 3z 2 56 z2 222 1 32 8 gt 15 5224z 2 3z8 gt 13 32228 gt 22103 gt y 173 andx43 General Method of Solving System of Linear Equations The example shown above is very simple and is not very general since it is not always possible to write the simple functions Instead we must use some rules of algebra to come up with a better method Adding Linear Equations If Eq1 and Eq2 are two linear equations that comprise a system of linear equations then mEq1 i nEq2 where n and m are real number multiples is also a member of the system of linear equations By doing this type of addition one can eliminate variables from the equations to get a set of new equations each consisting of only one of the variables example xy 3z2 gt 5x5y 152 10 5x y52 3 x 2y224 Ch45515 notes 1 sommer Linear Algebra By adding 5 times equation 1 to equation 2 we get 5x5y 15210 1 5x y 52 2 31 4y 102 7 Adding equation 1 to equation 2 gives x y 32 2 1 5x y 52 2 3 4x 22 2 1 We can continue this until we have manageable equations that can be used as they were in the original problem or we can go further and achieve the goal of getting three equations each of 1 variable m Our examples contained 3 equations and 3 unknowns Solutions to sets like that are said to be simple since only one unique set of values for x y and 2 work If there are MORE variables than equations the best we can hope for is a set of solutions that rely on one or more of the variables remaining a variable The quotknownquot values will be expressed as functions of the quotunknownquot variables This is the case with parallel lines Definition Matrix An array of numbers in an N by M grid corresponding to a system of linear equations Each cell in the grid corresponds to an element in one of the equations Let us use the generic linear equations Eq 12 a11x1 a12x2 a13x3 alej 2 b1 Eq 22 a21x1 a22x2 a23x3 aszj 2 b2 Eq k ak1X1 ak2x2 ak3x3 aijj bk The variables xj are multiplied by the coefficients akj where k is the counting number for the equation and j is the counting number for the variable Note If k gt j the system of equations is said to be quotover determinedquot while if k lt j it is quotunderdeterminedquot A matrix representation of the series of equations is for simplicity we have shrunk it down to 3 equations a11 a12 a13 b1 a21 a22 a23 b2 a31 a32 a33 b3 Each row represents one of the equations and each column except the last represents the coefficient of one of the variables The last column contains the unique solutions to the equations represented in the rows of the matrix Ch45515 notes 2 sommer Linear Algebra Definition Dimension of Matrix A matrix is said to be of MXN dimension if it has M rows and N columns Definition Row Echelon Form of the Matrix This is the form of the matrix where all the possible isolable variables are isolated In other words it is the form that reduces the number of variables in each row to one or more The first row in the echelon form uniquely contains variable 1 the second row contains variable 2 etc The row echelon form is obtained by performing Row Operations a Interchange Rows change the order in which the rows appear a11 a12 a13 b1 a21 a22 a23 b2 a21 a22 a23 b2 gt a11 a12 a13 b1 a31 a32 a33 b3 a31 a32 a33 b3 b Multiplication of Row by a Non zero Constant 321 322 323 a21 a22 a23 b2 2 a31 a32 a33 b3 can 09112 ca13 cb1 gt a31 a32 a33 b3 an a12 a13 b1 c Adding Constant Multiple of the Rows all a12 a13 bli ca111lt9121 ca12ka22 ca13ka23cb1kb2 a21 a22 a23 b2 gt a21 a22 a23 b2 a31 a32 a33 b3 a31 a32 a33 b3 Using combinations of these repeatedly we can transform a matrix into row echelon form example 1 13 2 4 4 12 8 1421 gtmultiplyfirstrowby4 14 2 1 2 0 4 3 2 0 4 3 5 0 10 7 gtadd second rowtop first 14 2 1 20 4 3 5 0 10 7 gtsubtract 25 of first row from last 14 2 1 00 8 15 5 0 10 7 gtdivide third rowby8 14 2 1 00 1 140 5 0 0 294 gtadd 10 times third rowto first 14 2 1 0 0 1 140 Ch45515 notes 3 sommer Linear Algebra 1 0 0 2920 gt divide first row by 5 1 4 2 1 0 0 1 140 1 0 0 2920 gt subtract first row from second 0 4 2 4920 0 0 1 140 1 0 0 2920 gt subtract 2 times third row from second 0 4 0 10040 0 0 1 140 1 0 0 2920 gtdivide secondrowby 4 0 1 0 1016 0 0 1 140 This last form can be read as x 2920 y 1016 z 140 Vectors as Row Matrices Vectors can be thought of as linear equations where each unit vector is the quotvariablequot Therefore Axiy3zhx y z This is referred to as a quotrow vectorquot note In the row vector notation the last column bk in the linear equations above is omitted Definition Transpose or Column Matrix In matrix notation we can view the vector A as a row as above or as a column This is referred to as the quottransposequot of the vector AT such that I x Ax y zand AT y 2 There is no inherent difference between the two notations Each serves a purpose Matrix Addition Two matrices of the same dimension may be added This is very similar to vector addition ie I a b cgh1 ag bh ci d e f 1 k l dj ek fl Scalar Multiplication of a Matrix A matrix may be multiplied by a number resulting in a matrix whose elements are each scaled by the multiple ie N a b c 2 Ng Nh N01 d e f Nj Nk N l 39 quot ion of Matrices Operators39 Initially when we introduced the matrix we referred to the elements in matrix as the coefficients of the variables in the system of linear equations We will improve upon that definition Definition Operator This is a matrix that acts as a function or mapping of a vector When the operator operates on the vector the result is also a vector Ch45515 notes 4 sommer Linear Algebra example Let us consider the system of equations Eq 1 a11x a12y a13z 2 b1 Eq 2 a21x a22y a23z 2 b2 Eq 3 a31x a32y a33z 2 b3 The vector that is common to all three equations is X x y 2 Each of the variables is multiplied by a different coefficient ajk depending on which equation it is in The array of variables is A a11 a12 a13 A a21 a22 a23 A a31 a3 3 where the symbol A is used to indicate that the matrix is an operator The result of the operation is the vector B 2 b1 b2 b3 How do we write the overall equation AXzB How do we perform the operation It turns out that when performing quotdotquot products on matrices the two matrices must share the quotinner dimensionquot Ie AMxN 39 BNXP CMxP Note 1 The dimension of the resultant matrix is the row dimension of the first matrix and the column dimension of the second Note 2 This is the origin of the term quotinner productquot In our example since the operator is a 3X3 matrix the vector it is to operate on must be a 3X1 matrix ie a column transpose vector Similarly the resulting vector must be a 3X1 matrix as well Ie I a11 a12 a13 K b1 a21 a22 a23 39YJ b2 a31 a32 a33 Z b3 Based on the original set of equations we can deduce that the multiplication process occurs when the corresponding elements in the first row of the operator matrix multiply the elements of the first column of the vector matrix to give the first element in the result matrix Similarly the second row of the operator multiplies the quotsecondquot column in the vector but since the vector only has 1 column the solitary column gets multiplies by the second row and subsequently the third row of the operator The result of each of these steps is the second and third element of the result matrix respectively Note 1 The matrix a11 a12 a13 b1 a21 a22 a23 b2 a31 A132 a33 b3 is referred to as the quotaugmentedquot form of A Ch45515 notes 5 sommer Linear Algebra If all bk 2 0 in the augmented matrix expression the matrix A is said to be quothomogeneousquot Note 2 Even though the quotdot productquot described in the vector algebra section of the course was invariant under commutation ie AB BA the inner product of matrices does not share this property However the transpose of the product does follow a commutation relationship namely ABTBTAT Scalar Product in Matrix Notation Knowing what we no know about matrix multiplication how can we obtain the same result we did for ordinary vectors Recall lxN row matrices OR le column matrices represent ordinary vectors Therefore we have two options when it comes to multiplying AlxN 39 Blle Clxl a scalar OR Al39le 39 leN DNXN a matrix When performing the dot product we expect to obtain a number ie a 1X1 matrix Therefore we would multiply a ROW vector by a COLUMN vector Definition Identity or Unity Matrix To insure that matrices follow normal rules of algebra there must be a matrix equivalent of the multiplicative identity ie the number 1 For square matrices ie those with the same number of rows as columns for example for 3X3 matrices the identity or unity matrix is A 100 II010 This matrix falls into a special class of matrices known as Diagonal Matrices since only elements along the diagonal are present We will see later that a diagonal matrix has special properties Other quotspecialquot matrices include quottriangular matricesquot where only the diagonal and elements above the diagonal appear and quotblock diagonalquot where the matrix looks like it is made of several smaller matrices all strung together along the diagonal Definition Inverse Matrix The matrix A has an inverse written A l such that A 1AAA 1I How do we determine the inverse of a Matrix There are two methods but we will only present one here The second will be described later Row Reduction Method This technique uses the ideas described when reducing matrices to row echelon form with a little twist The first step is to rewrite the NXN matrix as an NX2N matrix where the extra columns are made up of the NXN unit matrix elements In a 3X3 case this is written Ch45515 notes 6 sommer Linear Algebra A a21 a22 a23 0 1 0 a31 a32 a33 0 0 1 a21 a22 a23 a11 a12 a1310 0 a31 a32 a33 all a12 a13 This new matrix is allowed to undergo row operations until the first three columns represent the unit matrix The remaining 3 columns will then represent the inverse matrix a21 a22 a23 0 l 0 0 l 0 A21 A22 A23 all a12 a13100 100 A11 A12 A13 a31 a32 a33 0 0 1 0 0 1 A31 A32 A33 and all a12 a13 A11 A12 A13 1 0 0 a21 a22 a23 39 A21 A22 A23 0 l 0 a3 1 a3 2 a33 A3 1 A3 2 A33 0 0 l Definition Adjoint or Adjugate Matrix The adjoint of matrix A which is written Aadj is a special matrix where each element ajk in A is replaced by 1kjAkj in Aadj where Akj is quotthe determinant of the minorquot These terms will be define later but we will still use the adjunct concept now For a 3 by 3 matrix the adjoint is all a12 al3 I A11 A21 A31 A a21 a22 a23 and Ami A12 A22 A32 a3l a32 a33 A13 A23 A33 Note 1 The adjoint and transpose are NOT the same Note 2 A diagonal matrix is an example of a quotself adjoin quot matrix Definition Complex Conjugate Matrix If the matrix is quotcomplexquot it has both real and imaginary elements The complex conjugate of matrix A represented by A is the matrix where each i is replaced by i Definition Hermitian Matrix This type of matrix has the special property that the transpose of the complex conjugate matrix IS the original matrix A A r Hermitian matrices have very important symmetry properties that become very important in quantum mechanical theories Definition Unitary Matrix A matrix is said to be unitary if the transpose of the complex conjugate is the inverse matrix ie A 1 A r This type of symmetry becomes very important in Group Theory Definition Orthogonal Matrix This type of matrix has the property that the transpose IS the inverse Ail AT Ch45515 notes 7 sommer Linear Algebra which is also important in group theory Diagonalizing Matrices We have mentioned the notion that a diagonal matrix is self adjoint and we can also show that it is self transposed If all the elements in the diagonal matrix are real it is also Hermitian Now it remains for us to describe the process of diagonalization Coordinate Transformation We have already seen how we can transform xyz to R6 using certain geometric equivalences A similar process can be done to convert any arbitrary set of coordinates to a new more convenient set of coordinates Rotating the coordinate system to fit a particular case often does this In terms of matrix operations we can define a rotation matrix R that operates on a position vector X to give a rotated vector X39 The process of diagonalization involves finding the appropriate coordinate transformation rotation matrix such that R A R 1 A where A is diagonal The advantage of this is that the new matrix is equivalent in operation to the original but the number of actual operations multiplications and additions that need to be performed has been reduced from 9 to 3 in the 3X3 case Definition Eigen VectorEigen Value Equation There are special cases of operators that when they operate on a specific set of vectors the solution is just the vector multiplied by a constant The set of vectors and corresponding constants are referred to as quoteigen vectors and eigen valuesquot of the operator Symbolically this is written AszX In order to perform the multiplication by a vector we can notice that we can multiply the scalar A by the elements of a unit matrix of the appropriate dimension A A Kxzxix gt AX AIXA iX0 Therefore the matrix formed by the subtraction of the eigen value from the diagonal elements of the operator IS a homogeneous matrix This will be of great relevance in the next section and is of tremendous use in quantum mechanical calculations Definition Determinant of a Matrix A Denoted detA it is a quantity that may be obtained from a square matrix by using the following rule a11 a12 aln n det A det a21 a2 3quot a2n Z 1J39i ajidquJi 39 I 11 anl anz ann Z 1ri aji detAJi j1 Ch45515 notes 8 sommer Linear Algebra The difference between the two summations is that one is a sum across a row and the other is a sum down a column The term detAiJ is called a quotminorquot or a quotcofactorquot and is defined as the determinant of a matrix made by removing from the original matrix the column and row that contain the element aji Example The determinant of a 3X3 matrix a12 a13 an ax MA det a21 an an a11 12 det a32 a33 a31 a32 a33 a21 a23 a21 a22 a 13det a 14det 12 a31 a33 13 am an Note 1 Any row or column could have been used as the starting point for example if the second column was used instead of the first row the determinant would be a11 a12 a13 a a detA det am an a23 a12 13 det 21 23 a3l a33 a3l a32 a33 all al3 all al3 a 14det a 15det 22 a31 a33 32 an ax Note 2 Each of the 2X2 determinants need to be solved using the same technique For example a22 a23 detA11det m M 2 a22 14 det 833 a23 15 det 832 Note 3 The determinant of a 1X1 matrix ie a number is the value of the element For example deta33 a33 Note 4 You can always do row reduction on a matrix before taking its determinant By doing so you can generate matrix elements that are 0 which simplifies the overall calculation 3 D Vector Cross Product One of the most common uses of determinants is as mnemonic device to remember the cross product of 3 dimensional vectors The vectors are written as the second and third rows of a 3X3 matrix The first row is made up of the 3 unit vectors ie i Ajfk and the cross product is calculated by evaluating the determinant using the first row as the starting point In other words i 3 AXB det XA yA ZA 2 idet yA ZA jdetXA 2A E det XAyA YB ZB KB ZB KB YB XB YB ZB iQ AZB ZAYB j KAZB ZAXB E KAYB YAXB Ch45515 notes 9 sommer Linear Algebra Definition Singular Matrix A matrix is said to be singular if its determinant is 0 As we will see later singular matrices have no inverse Cramer39s Rule The determinant can be used to find the solutions to a system of N non homogeneous linear equations that have N variables The technique known as Cramer39s rule involves solving the determinant of the coefficient matrix and the determinant of a series of matrices constructed by replacing the variables column by the solutions An example will help illustrate the technique Example Lets look at the system of equations that satisfy the equation a1 1 a12 a13 K b1 a21 a22 a23 39 y b2 a31 a32 a33 Z b3 All the elements in the first column represent coefficients of x The solutions to the equations are XdetAx detAy ZdetAz det A y det A det A where a11 a12 a13 b1 a12 a13 det Adet a21 122 9123 det Axdet b2 9122 a23 a31 a32 a33 b3 a32 a33 a11 b1 a13 a11 a12 bli det Aydet a21 b2 9123 and det Azdet a21 a22 b2 a31 b3 a33 a31 a32 b3 Note This is not necessarily the most efficient method of solving the system of equations Inverse of a Sguare Matrix Previously we looked at the row reduction method of finding the inverse of matrices Now that we have the definition of a determinant there is another technique we can utilize The inverse of matrix A can be calculated using 71 1 Aadj det A This equality can be rearranged to give A Aadi det A i Solutions to Ei gen Value Ei gen Vector Problems We defined an ei gen value problem as one that could be written Kxzxix gt Kx xfxK xix0 Often times the form of the equation is known but the eigen values andor the ei gen vectors are not known In order to determine the ei gen values the determinant of the matrix A A i called quotsecularquot or quotcharacteristicquot determinant is set equal to 0 and the values of A that satisfy the equation ARE the eigen values The eigen vectors can Ch45515 notes 10 sommer Linear Algebra then be worked out by finding the correct vectors that satisfy the equation for each ei gen value Ch45515 notes 1 1 sommer Numeric Solutions to First Order ODE s the RungeKutta methods1 One of the simplest applications of a derivative is something called Euler s method It is basically a first order Taylor expansion of a function 3 09 E ymwoxxl x0 yxo y39XOAx With a good choice for Ax called a step size in most applications a very good approximation to the function at any point can be calculated if the function is known at an earlier point ie if we know yx0 an initial value or a boundary value For example if one wishes to compute the square root of 4 l we can use x0 4 as our starting point 1 1 01 4 z 447 41 4 272025 I F 214 4 A calculator showing 6 decimal places gives the answer as 2024846 which is pretty close to our estimate Of course the smaller the step size the better the approximation e g the square root of 4001 is identical using the calculator and Euler s method but we do not always have the luxury of having a small step size However we can use Euler s method successively on small steps to reach the target point In other words We can do the following 3 09 yxo yixoAx YOCZ z YX1 Y39x1Ax YX3 z YOCZ Y39x2Ax yx4 z yxn 41 y39xn 41Ax This type of sequence of operations is called an iterative process and each step is an iteration now we can see why Ax is called a step size To improve our results without adjusting the step size too much we can include more terms in the Taylor series A second order approximation would be yo yxo y39xoAx yquotx0Ax2 Using our example of the square root of 4 1 this second order approximation would give 1 1 1 1 1 2 01 001 If J4 2 1401 2 4 444 01 2 4 64 2024844 This result is nearly identical with the calculator value If we continue adding terms to the series we will eventually get the true value The biggest drawback to this technique is that we have to continually take derivatives evaluate them at the initial point and multiply by everincreasing powers of the step size Because of this another method has become very popular in the world of computational mathematics In this technique the higherorder terms in the Taylor series are approximated by calculating the first derivative at various points within the chosen step size Depending on how many terms 71 are included the method is referred to as a RungeKutta of order n in honor of the people who developed it There are variations on the main idea e g RungeKuttaGill but the basic approach is the same The approach stems from a simple first order differential equation of the form dy 7 F 4 dx x y Numerical ODE Solutions 1 mssommer We approximate the value of our function yx at a point between the known point yo yx0 and the target point yl yx1 using a simple approximation essentially the same as Euler s method yc0 AxZ z yx0 Fx0y0Ax2 yo k1h2 In this equation we have simplified the notation a little by replacing Ax with h and the initial value of the function F ie the derivative of y with k1 The derivative at the point x0h2 is further approximated by k2 Fx0 h2y0 k1 hZ This gets plugged into the Eulerlike equation for yl Y1 37061 yxoh yxok239h yo l39kz39h This equation represents the First Order RungeKutta Method The approximation for yl will improve as more kterms are added each corresponding to another derivative For example the Fourth Order RungeKutta solution is h y1 z yx0 gk1 2k2 2k3 k4 where the new terms are k3 Fx0 h2y0 k2 hZ k4 Fx0hy0k3h This is the method employed by MATLAB when the routine ode45 is called The syntax for calling the 43911 or 5m order RungeKutta routine in MATLAB is xy ode45yprime xO xfinal yO The symbol yprime is called a handle for the function representing the derivative of our function yx ie F xy in our derivation above This will appear in a separate mfile that has the name yprimem The vector x0 xfinal is the span ofthe variable x and y0 is the initial value of y The following set of commands written in mfiles will give you an idea of what to expect when using the ode45 routine xy ode45yprime 0 10 l ytrue xl plotxy39o39xytrue xlabel39x values39 function F yprimexy dydxFxy Fxy end In this example ytrue is actual solution to the differential equation included for comparison to the calculated solution marked with o s in the plot Numerical ODE Solutions 2 mssommer Calculus Review Definition Derivative The derivative of a function Fx is the instantaneous change of the function with respect to the instantaneous change in the variables Another way of saying this is that the derivative is the slope of a tangent line to the curve defined by the function Since the tangent line varies with the choice of variable the derivative is itself a function The derivative is given several symbols but the mathematical equation that defines it the so called limit definition is Fx 2 FM 2 hm w dX ma 0 AK Local Extrema Since the derivative is the slope of a tangent line the point when the derivative equals zero ie Fx 2 0 has a tangent line that is horizontal If the derivative at one of the points immediately to the side of the horizontal is a positive and at the other it is a negative the horizontal corresponds to a quotlocal extremumquot There are two possible extrema maximum and minimum Second Derivative The derivative of a function is itself a function The derivative of this new function is the quotsecond derivativequot of the initial function The second derivative describes how the slopes of tangent lines change as one moves from low x to high x from a given point 2 Fx FXXOQ d Fx Hm FxAx Fx Fx Fx Ax dx2 Ax 0 sz If the second derivative is negative then the function at the extremum is at a MAXIMUM Alternatively if the second derivative is positive the extremum is a MINIMUM If the second derivative is zero the point represents an INFLECTION POINT ie between a max and a min Definition The Differential of a Function The derivative is the change in the function divided by the change in the variable But if we wanted to know the amount by which the function changes as a result of changing the variable we are asking about the differential of the function dFx d Fx 2 dg dx This seemingly simple idea implies that the derivative IS a fraction Derivatives of Simple Functions Every derivative can be solved using the limit definition but the process tends to be tedious Several general results have been found that should be second nature Here is a list of functions whose derivatives should be known M2811an Mzanenx W dx K d W a n cosnx dam z a n sinnx dX dx Ch45515 notes 1 sommer Calculus Review Chain Rule Many functions are so called quotcomposite functionsquot meaning they are functions of functions which we can represent by GFx To take the derivative of G with respect to X requires us to first differentiate G with respect to F then differentiate F with respect to x This is called the chain rule and it is written d GFx d GFx d FX dx i d Fx dx In practice the function Fx is usually set equal to another variable say Fx 2 y This gives GFx Gy This allows us to write the more common form of the chain rule d Gy d Gy d y dx dy E Once again we can see that the derivatives are fractions Derivatives of Sums When two function are added the derivative of the sum is equal to the sum of the derivatives d Fx GX d Fx d Gx dx dx d X Derivatives of Products When two functions are multiplied the differentiation is done in the following way d FX Gxl GX Fm d Fx X dx This is usually called the product rule for differentiation d GX d X The g Quotient Rule A special case of the product rule is when the one of the functions is divided by the other ie FxGx To do this we will write the quotient as FX 1 F G GX X X This allows us to recognize that indeed this is a product Therefore the derivative is lexGxr1l lde dGxr1 dx Gx TX Fx dx What do we do about dGx 1dx This falls into the category of a function of a function Therefore we can write d Goorl dGxr1 d GOO 7 d GXl d K d Gx 1 G00 2 d X dx Thus the derivative of the quotient is lexGxgtr1l 1d Foo 2 lexl dx Gx TX Fx 1Gx1 TX GXW GX d goo RX d G00 Gx Fxx Foo Gxoo K d X GOQZ The last expression is known as the quotquotient rulequot although it is nothing more than the product rule with the chain rule Ch45515 notes 2 sommer Calculus Review Implicit Differentiation Not all functions are written in terms of y Fx Some functions for example the function describing a circle is a function of both x and y It is possible to differentiate the entire function with respect to x In doing so we will use the chain rule for the y yx dependency and solve for the exact form of dydx FXy x gy gt 3 x dx dy dx diy d FXy d foo d gy dx dx dx dy Example The equation of a circle is x2 y2 r2 where r is a constant the radius of the circle Taking the derivatives gives d x2 d yz d yz d y d y d r2 7 7 2 7 7 2 2 7 7 dx dx X dy dx X y dx dx Since r is a constant we can write it as rxO which when we take its derivative gives 0rx0 1 0 Thus we have d y d y d y 2x x 2 2 70 gt 2 72 2 gt 72 72 7 X y dx y dx X dx 2y y In other words it depends on both x and y Of course we could try to determine the function yyx and differentiate it but in most cases this is much more complicated Note 1 We can always find functions in terms of x and y and differentiate implicitly Note 2 We could have also differentiated with respect to y instead of x and arrive at dxdy implicitly But dxdy 1 dydx so the end result will have the same meaning Example Di ff 39 39 an inverse tr39 ic function Functions like arcsin sin l and arccos cos1 can be differentiated using the relationship dydx dxdy 1 This example demonstrates this for y arctanx y tan 1x gt tany tantan 1x x gt x tany dix d siny cosycosy siny siny cos2y sin2y dy leCOSQ cos2y cos2y sin2y 2 2 dy 1 1COS2y 1tany 1x ag 1X2 Definition Series A series represented by 22 is defined as the sum of a sequence of numbers that are somehow ordered in a set The ordering is a fairly quotloosequot restriction and it can be as simple as a collection of data points or as complicated as the set of results of some complicated function The only condition that is implied is that the elements of the sequence are discrete non continuous This will be clearer hen we do an example Ch45515 notes 3 sommer Calculus Review Finite versus Infinite Series Although the elements are discrete the sets can be of finite size or infinitely large An example of a finite series is the sum of the results of 10 experiments while an infinite series would be the sum of all positive integers Notice that the sum of all real numbers does NOT satisfy the discreteness condition Convergent versus Divergent Series Although we haven t yet mentioned any reason for considering series we can notice some simple properties of series One of these conditions is known as quotconvergencequot Simply put if the series converges it is equal to some number or simple function There are a couple of ways of talking about it The first is For a series 21 22 Z3 zm where zH are the elements of the sequence that comprise the series if lim zIn 0 In gt 00 the series converges and the series is convergent If not it diverges The second description known as the Cauchy convergence principle is a little more complicated A series 21 22 Z3 200 is said to be convergent if for some number of elements N there exists a value a gt 0 such that zn1 Zn2 zn3 zHP lt s where n gt N and p 21 Although this might seem like convoluted statement it is a more practical definition since it does not require one to consider an infinite number of elements Whichever definition you prefer you should always keep it in mind when considering series Ratio Test Another description of convergence requires that for some element zH in the sequence the ratio Zn1 lt 1 Mn If the ratio is gt1 the series d1verges Algebra of Series There are a couple of simple algebraic properties that need to be mentioned since they will be used again and again Multiplication by a constant 0 ZJZ CZj j j Addition of series leWjZjlZWjZZj Ch45515 notes 4 sommer Calculus Review It should also be obvious that 1N 1 H39Mz J A Common Series A common series that we will be refining shortly is the quotgeometricquot or quotpowerquot series This series is of the form 00 Z cszc020clzlc222m 3970 J The cjs are the coefficients of the variable 2 raised to its power The convergence of such a series depends on the value of z chosen as well as the nature of the coefficients A theorem that we will take advantage of is A power series can be used to represent an analytical function as long as the series remains convergent over the interval of the function Taylor and MacLaurin Series The uniqueness of a power series lies in the values of the coefficients and some central point where the value of the function is well defined This point is often 20 but is not neccessarily the case In general we would write 00 Fx 2 cJz aj c0 z a0 c1z a1 c2 z a2 0 where Fza 2 c0 This is usually referred to as expanding the function quotaround the point zaquot The values of cj may defined in terms of what are known as the quotTaylor coefficientsquot dJ39 Fz z a ie the th derivative of Fz evaluated at the point z a divided by j j factorial What is the 0Lh derivative The function itself In other words 1 dJ FZ Fa Fza Fzza FZ 0 l dzj inZ ay WJrT T This is generally called a Taylor series or a Taylor expansion When a 0 the series is usually called a MacLaurin series L Jj z a z a2 Some Common Taylor Expansions The use of Taylor expansions allows us to approximate functions that are otherwise too complicated to work with Computationally certain functions are exclusively evaluated using expansion Therefore it is worth explicitly showing some expansions dj sinx dxi sinx Z L x 0j sin0 cos0x J0J 0 sin0 2 cos0 3 2 X 3 X quot39 Ch45515 notes 5 sommer Calculus Review 3 5 7 0 1n 0 0L 0 L L 7 2nl X 3 5 7 nEben X cosx Z x 0Jcos0 sin0x wx2 an x3 F0 de 0 2 3 x2 x4 x6 1 2 1 0 E0E 0 a2 an n70 m1diex 3900 602603 mX ex 7 x 01 e e xixix 7 E0 dx P0 2 3 nEon 6139ng L Fol de 39 j 39 0 0 M x 01e0zer Lx2 1Lx3 P0 2 3 Z 1n 3 12 1n cosx isinx n O 39 n O 39 Note that the last example ei X is the result we used earlier during the complex number section Taylor Polynomials Truncated Series For many applications of the Taylor expansion only a finite number of elements are needed to get a good estimate of the function Stopping an infinite series before an infinite number of elements is called quottruncationquot The polynomial that is left after the truncation is called a quotTaylor Polynomialquot The number of elements presented in the polynomial defines what is called the quotdegreequot of the polynomial Periodic Functions Fourier Series Power series are not the best way to represent all functions Indeed periodic functions ones that have repeating units are poorly represented by such series Instead Fourier series may be used These are series of sines and cosines fZ a0 Z aj cos jz bj sin 02 39 1 J The coefficients aj and bj are called the Fourier coefficients and are also defined by integrals We will see these again later so we will not bother going over them now Diff quot quot of Multivariable Functions Many physical problems involve functions that depend on more than one variable Any 3 D phenomenon will require functions of at least 3 variables Therefore it is of value to consider what a derivative is in multiple dimensions Ch45515 notes 6 sommer Calculus Review Partial Derivative For example if we have a three dimensional function Fxyz we can as k how does it change with respect to one variable at a time Thus a simple concept is that of the quotpartial derivativequot which is the derivative with respect to only one of the variables The other variables are treated as independent of that variable In other words 8 Fxyz d Fxyz 8x 8x Y represents the change in the function with respect to x while ignoring the effect of y and 2 Since y and 2 can be any value the function generated by the partial derivative is a 3 D surface of all possible y and 2 values FxXyZ Multivariable Differential The differential of a function was defined earlier for a one variable case It represents the infinitesimal change in the function as the variable changes For multiple dimensions however it seems to be a little more complicated since each variable changes independently Therefore the total differential of a function of 3 dimensions is defined as d FogK d Fxyz dX d Fxyz dy d Fxyz dz l 8X fyz l 3y xz 92 xy Euler39s Cycle Relationship The three variables used in 3 D space ie x y and z are interdependent in most physical applications Usually one of the variables depends on the other two in an implicit way Therefore choosing 2 as the quotdependentquot variable we can represent it as a function of the other two i g dy 8x y By K We can take the derivative of the total differential with respect to one of the independent variables say x while holding z constant Z z zxy gt dz dx a 0ltz 6 Ly dxz 8xydxz dyxdxz 610agag ltLy dxz 8xydxz dyxdxz The partial of x with respect to x is 1 so this equation can be rewritten 82 82 By dz 8y 8x 7 7 7 gt 7 7 7 l dxy 8y dxz dyxdxzdzy The second equation is referred to as the cyclic relationship of the partial derivatives x Mixed Partial Derivatives When discussing derivatives at the beginning of the section the notion of a second derivative ie a derivative of a derivative made sense and it was attributed a special meaning curvature When it comes to partial derivatives the are two possible second partials the ordinary and the mixed The ordinary second partial is for example 82 Fxyz 8 8x2 8X 8 Fxyz 8x The mixed partial as the name implies involves a mixture of variables For example the mixed partial of Fxyz with respect to x and y is Ch45515 notes 7 sommer Calculus Review 82 Fxyz d d Fxyz OR 82 Fxyz d d Fxyz dxdy dxl 8y dydx dyl 8x The two possibilities are only equal when the function and the first derivatives are continuous Gradient Divergence and Curl The total differential of a 3 D function involves each of the three first partial derivatives The question arises as to what it represents graphically To explain this we must introduce the concept of the quotgradientquot symbolized by an upside down delta called a quotdelquot V and is defined as which when operated on a function Fxyz gives a vector More specifically the resulting vector called the gradient of the function is in the direction of MAXIMUM INCREASE in the value of F The total differential is obtained when the gradient is quotdottedquot with a vector representing the change in the variables dF VFx y zdX Bigbx dy dz 8x By 82 The gradient is only defined for scalar quantities since the multiplication of the operator with the function is not defined in terms of a quotdotquot or a quotcrossquot product When the quotdelquot operator is used with vector quantities the choice of dot or cross makes a big difference The quotdivergencequot is defined as the dot product of a vector function with the del operator resulting in a scalar The scalar represents the quotfluxquot ie out ow minus inflow of the vector through the point at which the divergence is evaluated The quotcurlquot is the cross product between the del operator and the vector function Clearly the result is a vector that represents as the name implies a rotation More specifically in the case of a velocity the curl of the velocity represents a vector pointing in the direction of the axis of rotation This is has many consequences in fluid dynamics and electromagnetism Laplacian Operator The second derivative of scalar function in 3 D involves taking the divergence of th gradient of the function VVFXyZ V2FXyz 82Fxyz 82Fxyz 82Fxyz 8x2 By2 822 This is of course also a scalar The operator V2 read quotdel squaredquot is called the Laplacian operator Inverse Differentiation For a function fx there exists a function Fx such that d Fx 7 fx dx The function Fx is the inverse derivative of fx Area Under the Curve and Riemann Sums Definite Integrals The derivative above can be rewritten in terms of the total differential Ch45515 notes 8 sommer Calculus Review dFx fx dx At a given point xx0 this representation can be understood as the value of the function y fx at x x0 multiplied an infinitesimal change in x ie dx Ax Since Ax is a length along the x axis and yo fx0 is a height the product corresponds to the area of a rectangle under the curve at point x0 We can add up all of the rectangles defined by fxiAxi for all points xi along the curve ie Z fXiAXi This sum would correspond to the total1 area occupied by all the rectangles In the limit of Ax gt 0 the rectangles would be infinitely narrow and the sum would correspond to the area bounded by the x axis and the curve y fx n xquot lim Z fxiAxi 2 Area fx dx AX T O i 1 x1 The sum is known as a Riemann sum and the symbol at the end of equivalence is called a definite integral where x1 and xH are the limits of integration Fundamental Theorem of Calculus We can relate the Riemann sum back to the original inverse derivative using the limit definition of the derivative lim FxiAxi Fxi lim fxiAxi Ax a 0 Ax a 0 Summing this over all the values of x from x1 to xH gives x H H fx dx lim 2 fxiAxi lim 2 FxiAxi Fxi x1 Axao i1 Axaoi1 In the last sum the term Axi xi1 xi so that the sum can be rewritten H H 2 Fx1x11 xo Foo Z lFXi1 Foo i 1 i 1 Since the sum uses all Fxi in both the and forms all except x1 and xml the sum comes down to xn n f fX dX lim 2 IFX11 FXil FXn 1 FX1 Ax a 0 i 1 Since Ax is so small Fxn1 is almost the same as Fxn which gives us the expression I fx dx 1 dFx Fxn Fx1 x1 X1 This expression is called the fundamental theorem of calculus Indefinite Integrals The Riemann sum requires a specified range of values for the integral for example xH s x 5 x1 However just as derivatives can be viewed generally not just at a specific point on a curve the inverse derivative or integral can also be generalized Thus by omitting the limits of integration we get Ch45515 notes 9 sommer Calculus Review fx dx Fx C where C is the constant of integration The C is required since we know that d FX d7 idx fx and dX FX C fx Integrals of Common Functions In our discussion of derivatives we came across several functions that were so common that the intimate knowledge of their derivatives was necessary For integrals there are several functions that are so common that we need to become comfortable with them kxndxkixn1C foralln 1 n 1 fkidpkzmxnc fekxdleiekxc f sinkx dx coskx C I coskx dx E sinkx C Odd and Even Functions Some functions are symmetric with respect to rotation about the y axis ie fx f x For example cosx takes on the same value as cos x These functions are called even Other functions for example sinx are antisymmetric with respect to the rotation ie fx0 f x These are called odd functions Recognizing the oddness or evenness of a function often simplifies the definite integration process 8 8 I fevenx dX 2 fevenX dx 0 7a a J foddX dX 0 7a Recognizing oddness and evenness is a great time saver Integrals of C quot J Functions T 39 of I quot bv Parts and qllbstitntinn39 The derivative of a product of functions requires the use of the product rule A d gx d fx dx WK gXl fX dx 00 dX The integral therefore would be dfxgx J fx 5 dx goo i X dx fx d goo I goo d fx dx Therefore recognizing an integrand as fx d gx allows us to write fX d 00 dlfX39gXl gX d fX lfX39gXl gX d fX The second integral integrand gx d fx may be simpler to deal with thus making the integral soluble This technique is known as integration by parts Ch45515 notes 10 sommer Calculus Review Similarly when we took derivatives of composite functions we had to use the chain rule A f 2 d fx d goo dx 1 gxl dgx dx Integration requires d fgX d gX d fgX d1fgxl J dgx dx dX dgx d gx This technique is known as the method of substitution and is used when we can recognize the integrand as a composite function Other Integration Technigues There are several mathematical tricks that allow us to integrate functions that on the surface seem too difficult Several of these techniques are covered in the text and are done so with good examples PLEASE look over these sections as we will be using these techniques where necessary Integrals as Area Part II Since the integral is the are under the curve it stands to reason that the difference between two integrals is the area between the two curves fx dx gx dx Fx Cf Gx Cg It also makes sense that the order of subtraction and integration is not important ie J fx dx I goo dx 1 fx goo dx Example Volume of Rotation and Surface of Revolution The area is measured in units of length2 but if the function is rotated about the x axis a 3 D shape is traced To obtain the volume of this shape we can recognize that we are including all values of y fx and y fx plus the z fx and z fx and all points in between This is basically a sum of discs or radius r fx centered at each value of x The area of each disc is m2 nfx2 and the sum over all discs is Volume 1 7 fx2 dx The surface area of the 3 D shape is obtained by summing the circumferences of the discs ie summing 2m 2 2nlfxl The infinitesimal width of the disc ie dx is no longer the thing we are summing ie integrating over since we are not interested in the whole disc just the surface Therefore we sum over the infinitesimal change in the length of the surface dl We can approximate the surface as the hypotenuse of a triangle whose width is Ax dx and whose height is Ay dy dfx ie 2 d1 dx2 dfx2 dx2 1 7 2 dx Putting this all together gives us the formula I 2 Surface Area zj 2 339 fx 1 dng dx Ch45515 notes 1 1 sommer dfx 2 dx dx 1 Calculus Review What if we rotate about the y axis instead The area being measured is no longer m2 rather it is 2nxy 2nxfx where 275x is the circumference and fx is the height The integral for the volume becomes Volume zf 2 aquot x fx dx Averages Using Integrals Clearly the summation quality of the integral allows us to think of integrals as totals Therefore it is only natural to think of uses for the quottotalquot value of a function One such application is in the field of statistics and specifically the concept of averages and means When we consider an average value of a function what we are actually thinking about is some value the function takes over a range of variable values For example if a function fx is examined over a range of x values the average value would be fx dx Average dx ie the sum of the values of fx multiplied by the values of x divided by the sum of values of x In common usage the function may be a student39s grade while the sum of x values is the number of students In this case it is often useful to introduce the concept of a distribution function which describes the number of occurrences of a particular value of fx and which sums up to equal total number of all occurrences The distribution function which we will call gx therefore can be used to find the average fx gx dx Average gx dx A normalized distribution is one where the function gx is divided by the total number of occurrences say N which would give gill dX gnormx dX 1 The normalized distribution function represents the fractional occurrences of each value of the functions The most commonly used distribution function in physical sciences is the so called Boltzmann distribution function eiax eiax gnormX 00 N e x dx 0 where the variable x and the coefficient a have to do with energies and temperatures The Boltzmann distribution is used to find average values of physically relevant properties that are functions of x yielding expressions of the form Ch45515 notes 12 sommer Calculus Review 00 fxe ax dx 0 I N where e xdx N is the fraction of occurrences of the value fx in the interval between X and xdx Integrals Involving Vectors Multivariable I Functions of more than one variable can be differentiated more than once and with more than one variable Therefore it is important to know how to perform inverse differentiation integration on such functions Depending on the physical significance of a function integrating it will take on more than one meaning Line Integration The total differential of a function as you recall represents the change in the function as each variable is varied This was further described in terms of a gradient vector and a differential distance vector F c d F 1 d F A dF VFxyzdX 1 7 E k dx dy dz 8x 8y 8F 8F 8F dF dewdygdz Therefore when we integrate dF we are actually performing integrations of three different functions the partial derivatives each with respect to a different variable 8 F d F d F dF 7 d 7 d 7 d dx X By y dz Z This type of integration is referred to Line Integration since we are defining a line dxi dij dzAk over which we are summing Multiple Integrals The mixed second partial derivative of a two dimensional function has importance in describing a function as continuous Therefore quotundoingquot the two derivatives could also be important This is done by integrating the function twice once with respect to dx and the second with respect to dy f f Gxy dx dy f f Gxy dy dx The order of integration should not matter just as the order of differentiating did not matter This is referred to as a double integral The idea can be extended to 3 D functions ie triple integrals and so on The volume of rotation of a 1 D function can be recast in terms of a multiple integral of a 3 D function The term dxdydz is in the same units as a volume ie length cubed therefore it represents the volume of a cube of dimensions dx by dy by dz Since not all volumes of interest are cubes we can use the same idea we used in the integral description of the surface area Instead of generic dx dy and dz we will use differential distances along the quot generating functionsquot of the volume For a spherical volume the volume element is r2sin6drd6d The general conversion from one set of coordinates to another is called a quotJacobian Transformquot and is given by the formula Ch45515 notes 13 sommer Calculus Review 8x 8x 8x 3 W J det 81 81 al such that dxdydz J dudvdw du 8v 8w 82 82 82 E E W Therefore any coordinate system can be used for the integration Application Gaussian Integrals A really neat application of the multiple integralchange of variable procedures is in determining the value of the following definite integral 00 Izj xn eraxz dx n 0 1 2 m This integral is called a Gaussian integral since the eraquot2 term is in the form of a Gaussian distribution This type of integral appears often in the physical sciences and thus is an important one to examine To determine this integral we must start by examining the case when n0 ie 00 I zj eraxz dx To evaluate this integral we will first consider the square of the integral 00 00 00 00 I2 11 J e m z dx I e y2 dy 1 I e m z e y2 dx dy We are allowed to use the different variables x and y since each of the definite integrals will yield the same value since the limits of integration are identical We can rewrite the integrand as 00 00 I2 I I e W Zt yz dx dy We can recognize that x2y2r2 and we can change dxdy to rdrd6 giving 00 00 2n 00 I2 I I e41x2y2 dx dy I I e419 r dr d6 40 m 0 0 The limits of integration have changed to reflect the new variables Since the integrand is not an explicit function of 6 we can perform the de integration first to yield 00 00 I2 2339 e413 r dr 2 aquot e mz 2r dr 0 0 The term 2rdr is nothing more than dr2 so we can write Ch45515 notes 14 sommer Calculus Review 00 00 I2 2 TI e413 2r dr 2 TI e413 dr2 0 0 The variable of integration is r2 and the variable in the exponent is also r2 so we can use a well known integral feaxdx 1a e to solve this 00 100 2 7 2 2 E r2 E I n eardr ae41a 0 r0 Since this is 12 all that is left to do is to take the square root to find 1 00 12W e m zdxz 700 However we started off this section by mentioning that Gaussian integrals are of the type 00 1 xn e m Z dx n0 12 m and meanwhile we only have found the integral when n0 What about the cases when n 0 To do those we need to recognize two facts 1 when nodd the product of x e m 2 is an odd function which integrates to 0 over the range 00ltxlt00 and 2 If we take the derivative of ed 2 with respect to a we get x2e ax The first quotfactquot allows us to do very simple calculations IF the limits of integration are indeed 00 to 00 As we will shortly see this is not true for integration between 0ltxlt00 The second quotfactquot can be extended to any EVEN power of x Specifically the second derivative of ed 2 with respect to a is x4e x2 etc Therefore we can write 8a 2n VE 1 n 22H anl2 00 2 n TI 2 x211e ax dxziidn 7 da 40 We also know that an even function integrated from 00 to 00 is the same as twice the integral from 0 to 00 ie 00 00 x2n e m z dx x2n e m z dx 0 700 2n V n 22H1anl2 The textbook goes into some detail about how to integrate odd powers of x from 0 to 00 so we won39t go into those details Instead we shall just give the results here 00 X2nl eiax2 dx O 2 an l Henceforth we can use these results without ever having to derive them again Ch45515 notes 15 sommer Differential Equations Differential Equation A differential equation is an equation that includes at least one derivative of an unknown function These equations may include the unknown function as well as known functions of the same variable The derivative may be of any order and there may be several derivatives present Generally speaking a differential equation is a representation of a physical phenomenon where the derivatives correspond to the quotrates of changequot of the unknown function with respect to the variable The solution to a differential equation is the form of the unknown function that makes the differential equation into an identity Classification of DiffEgs There are many terms used to describe a differential equation The first is the order of the differential equation which describes the highest order derivative in the equation For example if we had the equation 137 x 1 x2 diy dx3 X2 dx the highest order derivative is d3dx3 which makes this a quotthird orderedquot diff eq It is often advantageous to rewrite the differential equation such that the highest order derivative is isolated In our example this becomes d3y d2y d1 d x3y0 7 xix2 2 x x In other words the highest order derivative is a function of the variable in this case x and all of the lower ordered derivatives X3yl FX y y yquot Another important classification is that of linearity If the unknown function and its derivatives are present to the first power ONLY the diff eq is said to be linear If any of them are present raised to a higher power the diff eq is nonlinear Note the independent variable x may be present in any power without affecting the linearity The degree of a differential equation is the power to which the highest ordered derivative is raised The last classification of a diff eq is that of being ordinary or partial If the derivatives are NOT partial derivatives the diff eq is said to be ordinary Conversely if partial derivatives are present in the equation it is said to be a partial differential equation This distinction will become more obvious when we look at how to solve these equations Solving Diff Egs Finding the differential equation that quotmodelsquot the physical phenomenon is probably the most important problem However the solution to the diff eq ie the function ie a relationship between the variables that satisfies the equation is what we will be looking at in this section To simplify the discussion we will look at the different classes of diff eqs separately starting with the simplest and building up to the more difficult Solving diff eqs requires some expertise in integration as well as a good deal of intuition Recognizing patterns often simplifies the work of solving the diff eq First Order Ordinarv Diff Eqs ODEs First order differential equations fall into several categories They are given by the following equations Ch45515 notes 1 sommer Differential Equations dy Ly 1 dX fx 2 dX gy d d 3 fXgy 4 i hXy Each of these is slightly different thus requires a slightly different technique to solve We will look at each one individually dy 1 7 function 1e y fx dx Since we have spent much time talking about integration we will not spend any more time discussing this class of diff eqs fx This type of differential equation can be solved by direct integration of the 2 2 gy Although this type of diff eq seems much more complicated than the previous one it is very similar The big difference is that the role of the two variables X and y have switched The variable y is now the quotindependentquot one and X is the quotdependentquot one To solve this we rewrite the equation to reflect the new classifications ie dy d 1 7 gt7X hltgtx hd dX gy dy gy y y yy Once the equation of X as a function of y is generated one can invert the classification again and generate y yx 3 3 fxgy This is the first really quotinterestingquot type of diff eq that we will see Solutions of this type of diff eq require us to recognize that the equation can be rewritten as dy 1 ifx gt fXdXid gt fxdxG d 0 dX g y gy y y y where Fx 2 1fx and Gy 1gy This last form indicates that the two variables can be treated separately thus this technique is called quotseparation of variablesquot The solution to the equation will be of the form f d d box gyy Once the two integrals are evaluated a functional form of y yx may be deduced This type of differential equation is also related to the form dy fxy dX gxy where the functions are not of just one variable Solutions to these types of equations require some of the techniques used in the line integral section of the course Often a substitution of variables e g y fxz is required to allow for the proper solution Examples of this will be given in the next section gt fxy dx gxydy 0 Ch45515 notes 2 sommer Differential Equations 4 2 hxy This type of differential equation has several subcategories The first ones which we have looked at were described in the description of quottype 3quot The other ones will be discussed separately below i Ax 3 Bx y Cx This type of differential equation is known as a quotLinear equation of order onequot Clearly it can be rewritten so as to isolate the derivative dy 1300 C00 dy 7 7 F G dx Ax y Ax gt dx X y X Once it is in this form it can rendered into the more familiar form of quottype 3quot which we can solve more easily The first step will be to rewrite the equation as a total differential 3 Fx y Gx gt dy Fx y Gx dx 0 To solve this we will introduce the concept of an INTEGRATING FACTOR which we will denote vx This factor when multiplied by the total differential form ie vx dy vx Fx y Gx dx 0 makes the total differential exact ie 6H 6H 82H 82H dHxy a dy 3 dx and axay 9an In this example 8H 9H W vx and a vxyFX Gx Plugging into the condition for exactness ie mixed second partials being equal gives 82H 82H d vx a m 9an gt dX E VXyFX GX VXFX This can be rewritten d vx dx vxFx gt m Fx dx gt m Fx dx vx VX The first of these integrals is just the natural logarithm of v which leads to lnvx Fxdx gt vx emxmx This is the integrating factor for this equation What remains to be shown is how we use it The first step is to plug it back into the exact total differential e xdx dy emxmx Fx y dx emxmx Gx dx It is clear that if the function Hxy were ye xdx then the left hand side would represent the total differential Thus we can conclude that if we divide by dx we get etFltxgtdx 1 eLFltxgtdx Fx y etFltxgtdx Gx gt jib etFltxgtdx eLFltxgtdx Gx x x Since the right hand side is JUST a function of x we can integrate as normal d y emmdx emxmx Gx dx gt y emxmx emxmx Gx dx Once solved we can divide by the integrating factor to obtain a function for y yx Ch45515 notes 3 sommer Differential Equations ii Ax 2 Bx y Cx yn This subcategory of equations falls into the category of Bernoulli39s equation As with the previous class of differential equations the first step is to rewrite the equation in a more convenient manner we 3 1300 y Cx yn gt ym 3 Foo yin mm In this form it is easy to see that a substitution can be made namely 2 V 1 By doing this we can see that z V11 gt dz n1y n dy This allows us to rewrite the diff eq as follows dy 1 d H 7 F Ht1 2 G gt i F G V dX X y X 1n dX X Z X This resulting equation can be solved using the method of integrating factors Linear Combinations of Solutions In several cases there exist more than one solution to a differential equation For example let y1 and y2 both be solutions to a diff eq In such cases any linear combination of the solutions is also a solution ie y3 ayl byz where a and b are constants is also a solution This will be true for all diff eqs that are said to be quotlinearquot Boundary Valued Problems Most of the solutions we have seen include some sort of constant of integration This is necessary since the we are performing indefinite integration which necessarily has a constant associated with it However the physical problems that are represented by the dif eqs do not include an arbitrary constant Rather only specific choices of the constants have physical meaning For example if yx represents some physical quantity like position which depends on time then for a particular case only the positions encountered during that experiment are valid In general the initial position is predetermined by the experimental set up so that yx0 2 yo is a definite solution to the problem This type of known solution is known as a quotboundary valuequot and any functional form of yx derived from the diff eq MUST satisfy the condition y0 2 yo This allows one to determine the ONLY valid constants of integration Second Order ODEs Many physical phenomena are related to derivatives of a higher order than 1 For example forces are related to acceleration which is the second derivative of the position with respect to time This section will be devoted to second order diff eqs ie those whose highest derivative is the second derivative Linear Second Order ODE The simplest form of a second order diff eq is the so called linear equation which is of the form 2 mo 37 goo 3 hx y kx This is linear in the order of the derivative where the variable y is the zeroth derivative Note if the function kx is identically zero the equation is said to be quothomogeneousquot as well as linear Solutions to these equations will be more complicated than the first order ODEs but we will look at several cases to see how to go about solving them Before we Ch45515 notes 4 sommer Differential Equations do so however it should be pointed out that linear combinations of solutions to these equations are also solutions a General Solution of l Linear Second Order ODE Tavlor Series Recall that a homogeneous equation is of the form dzy dy RX 172 gx E hx y 0 We know that a solution y yx can be found and by Taylor s theorem we can generally write 2 711237 n O Plugging the Taylor series into the diff eq and taking the derivatives we get d ya x a 2 dnya x ar 1 dnya x an f 7 7 h 7 7 0 bongo dxn n 2 goongo dxn n l XLEO dxn 11 Since the different powers of the series are independent then the only way this equation can equal zero is if the coefficients of each power is zero This amounts to 2 fx W gX M hx ya 0 for X a0 dx2 dX 3 2 fx W gX W hx dig 0 for x a1 and so on Thus we can determine the Taylor series form of the function as long as the values of ya and y39a are known These two values are the boundary conditions for the solutions to this second order diff eq b l with Constant Coefficients39 The simplest linear second order ODE is one where fx f gx g and hx h where f g and h are constants Of course if some of these constants are zeroes the problem becomes even easier As with the general linear equation we will first look at the homogenous equation de7y d7 h 0 2 8 dx y A simple solution to this equation would be one where each derivative return the function and a constant d7y my and 127 m2y dX dx2 A function that fits this is the exponential function y emx This leads to what is called the quotauxiliary equationquot fm2ygmyhy0 gtfm2gmh0 The last equation is a quadratic equation which can be solved using the quadratic formula 2 f Thus the two simple solutions are y eHm and y eIILX Plus the linear combinations Ch45515 notes 5 sommer Differential Equations y cy cy cemx Lem It should be pointed out that this technique is true for higher order linear homogeneous ODEs where higher roots are required Although these are not as easy to find eg the cubic formula is not as simple as the quadratic formula they can be obtained iteratively using a computer b i Repeated Roots In the event that rm 2 UL a case known as a repeated root then the two simple solutions appear to be identical and the linear combination is just a constant multiple of the solution This is NOT a viable answer Instead the two solutions are y1 eInx and y2 x eInx and the linear combinations are y CIY1 c2 2 016mx 02 X en To verify that y2 is a solution we can plug it back into the original diff eq deiy g 11 h y f dZX emx g LX emx h x emx 0 dxz dx dxz dx Doing the derivatives we get f 2m eInx m2xeInx g eInx m x emx h x emx 0 Dividing through by emx gives f2m m2xg1mxhx2mfgm2mghx0 In order for the two roots of the auxiliary equation to be equal it must be true that gW g Wm 2 f 2 f This occurs when vg2 4fhzqLeg24thNDn1Z Plugging this result into the diff eq for y2 gives x0 2 2 i i i 2mfgm mghx 22ffg 2f 2fgh multiplying through gives 4f 2f4fx0 gm thus verifying the solution b ii Complex Roots Another quotcommonquot example of a second order ODE with constant coefficients is one that yields complex or imaginary roots for the auxiliary equation Generally speaking the two roots are nnaib and nnza ib These yield the two solutions yJr emu Ca ibx eaxei bx and yi eaxei i bx In both cases the eax term is equivalent to the emx term above The eii bx term is one we already recognize as being et 3quot cosbx i i sinbx These two solutions can be combined to give the general linear combination result y c1y c2y c3eaxcosbx i c4eaxsinbx where c3 c1 c2 and c4 2 c1 c2 Ch45515 notes 6 sommer Differential Equations c N n Second Order Linear ODE Now that we have a better feel for second order diff eqs we can make the leap from homogeneous to non homogeneous equations The general form of course is d2y dy fx Q gx amp hx y kx The related homogenous equation ie when kx 0 is called the quotcomplementaryquot equation The solution to the complementary equation is symbolized by ya and we have briefly talked about it ie the Taylor series solution The general solution to the non homogenous equation will necessarily include yc as well as a quotparticularquot solution yp which is one solution that definitely works We will look at these solutions shortly Simply said the general solution will be y yc yp This can be verified by plugging into the diff eq fX gX d2 d y 2yp yaxyp MK ycyp K d2 d d2 my goo hx y mo 1 Now that we are satisfied that y yc yP is a solution we will need to examine the particular solutions d gx hx ypl 0 kx c i Method of Undetermined Coefficients This technique can be used for diff eqs with constant coefficients ie dzy dy fdizgghy kx Note the non homogeneity ie kx need NOT be a constant Of course we know the solution to the complimentary equation ya from our discussion above The particular solution is what we are interested in To find it we must treat the function kx as a particular solution to some OTHER homogeneous linear equation with constant coefficients This may be of higher order than 2 ie aLkOQ b Lilkog w dk dxn dxnil dx Using the technique of auxiliary equations we can generate the necessary roots m zkx0 am nbm n 1wm z0 F1nd1ng this equation is the key to solv1ng the problem Once this differential equation is found we can apply it to the non homogeneous equation of interest 2 2 2 addfd7 g7hy wfd7 g7hyzfggiihy 0 Clearly this can be rewritten as a linear ODE with constant coefficients of order n2 which will have n2 root Inquot These roots will contain the original root In of the second order ODE and the roots m39 of the kx equation The general equation of the kx equation can be adjusted to fit the particular non homogeneous equation we are trying to solve And if we have an initial condition the exact form of the complementary solution can also be found c ii J quot of Order and Variation of P The 39 1 above only works when kx IS a particular solution to a diff eq AND we can find the diff eq However Ch45515 notes 7 sommer Differential Equations this is not always the case Occasionally this is not possible and another approach must be employed One such technique called the quotreduction of orderquot is similar to the method of integrating factor used when discussing first order ODEs Once again we are dealing with constant coefficients ie dzy dy fgg hy kx with the complimentary equation d2y dy 7 7 h 0 f X2 g dX y However this technique is not RESTRICTED to constant coefficients The general idea in this method is that a solution to the non homogeneous equation is a solution to the homogeneous equation which we will call yh multiplied by another variable V vx ie y yhv Plugging into the original diff eq gives d2V39Yh dV yh d d dyh 7 h 7 7V 7 f M g dx vyh fdX YthV dX a in dxV dx gyh hvyh d7v dy7h gyhdxvdX hvyh d2v dv th dZYh 7 277 7 f yh dx2 dX dX V dx2 2 d d2 d Zflyhf l x dx22 amp d7vd d yh in f yhdxz 2dx dx fdxz g dx hyh f amp a in g M1de 2dx dx gyhdX VlOl kx In other words we have a new differential equation in terms of vquot and v39 Yh d7d7v thdX2gYh2de dx kX gyh 7v By letting w v39 we can change this from a second order diff eq into a first order ODE thi gYh 2fwkx This can be solved using an integration factor A related technique known as quotvariation of parametersquot relies on knowledge of the general solution to the complimentary equation yc c1y1 c2y2 Whereas the method of reduction of order only uses one of the solutions this method uses both The general solution to the non homogeneous equation is taken as y Axy1 Bxy2 ie the constants are replaced by functions of x Therefore the derivative of y is all Cm d3 WK dBltXgt AX d 13XdX Y1 dx 2 dx To make the choice of Ax and Bx simpler we require that X dBX 7 0 dx Y1 This allows us to write dzy d2Y1 1232 dy1 dAx de dBx 7 A 7 B 7 7 7 dx2 X dx2 X dx2 dx dx dx dx which when put into the original diff eq and rearranged gives the equation Ch45515 notes 8 sommer Differential Equations dAix d dBOQ MK We now have two equations with two unknowns A39x and B39x so e can solve for them to obtain the general solution d Method of Laplace Transforms A transform is a special type of linear operator that changes the variable of a function to a different one via the following integral equation 00 TFt zj Kst Ft dt fs The function Kst known as the quotkernelquot of the transformation effectively transforms the function Ft into the new function fs The choice of kernel depends on the application In particular we will be interested in the Laplace kernel which is defined by 0 for t lt 0 Kst e st for t 2 0 This definition gives us the Laplace transform oo LFt 1 erst Ft dt fs 0 The limits of integration are 0 s tlt 00 since for t lt 0 the functional form of Kst is 0 The usefulness of Laplace transforms in solving diff eqs is in the simplicity of the resulting functions Specifically lets look at the Laplace transform of the derivative of YO dy m 7 dy 7 00 m 7 ME 1 e S Edt yte stltO s1 e stya dt The last integral is just Lyt and applying the limits of integration to the first term on the right gives d mg ylto s Lyltt Similarly if we replace the first derivative with the second derivative we get d2y dy L 7 0 L 7 dtz ylt s dt which upon substitution of the Laplace transform on the right gives d2 M35 yro s yo s Lytgt1 s2 Lyt s W yro In other words as long as we know the initial values of y and y we can find the transforms of the first and second derivatives Applying this to a second order diff eq with constant coefficients gives d2y dy dzy dy fj gThykX gt fLig L hLyLkX X X X2 dx 2 fls2 Ly s yo y390l g ym s Ly1 h Ly Lkx Rearranging this equation to isolate the Ly terms gives Ch45515 notes 9 sommer Differential Equations L kx s 0 0 My 2 g y fy fs2 gs h To find y all that remains is taking the quotinverse transformquot of this solution Using a table of Laplace transforms and finding the one that best fits this solution accomplishes this Ch45515 notes 10 sommer Complex Algebra Imagina Numbers Multiples of V l i Properties ofiii i2 2 1 and i i ii 1 Complex Numbers C composed of a real number R and an imaginary number For complex number Z X iY where X and Y are real numbers we can define the quotoperatorsquot ReZ X and ImZ Y Agrand Diagram The real component of a complex number is taken as the x component of an ordered pair and the imaginary part is the y component The complex number is the point on the graph corresponding to xy y Imz xoyo ZO x Rez Algebra of Complex Numbers For most of the discussion we will use two generic complex numbers zxiy and waib Addition and Subtraction z w xiy aib xa iyb z w xiy aib x a iy b Multiplication z 0 w xiyaib xaib iyaib xa ixb iya yb xa yb i xbya z 0 z 22 xiyxiy x2 y2 ix2y2 xZ1i yZi l Definition Complex Conjugate If ZXiY is a complex number then its complex conjugate 2 is 2 X iY ie i is replaced by i z 0 2 xiyx iy x2 ixy iyx y2 x2 y2 A real number NOTE ZW ZW Ch45515 notes 1 sommer Complex Algebra Definition Modulus The modulus of a complex number IZI X1Y is given by 2 zz 2 llx2 Y2 NOTE The modulus is the distance to point z from the origin in an Agrand diagram Division A l W zw xiya ib xayb ixb ya xayb iya xb W W w w2 a2 b2 a2 b2 a2 b2 NOTE This is the same idea as rationalizing a denominator Fundamental Theorem of Algebra A complex polynomial of order n has exactly 11 complex roots some of which are NON unique Polar inn of Complex Numbers Especially useful when thinking in terms of Agrand diagrams v mm 1 V I x0y0z0r060 U r I 0 1 quot68 xRez Based on trigonometry x r cos6 and y r sin6 z x iy r cos6 i sin6 Ill 2 r2cos26 sin2612 r Definition Argument Arg The argument of a complex number ArgZ is the angle that represents it in the polar representation of the number ArgZ ArgXiY ArgRcos6isin6 6 Review Trigonometric Identities cos oc cosoc EVEN FUNCTION sin oc sinoc ODD FUNCTION sinoc5 sinoccos5 cosocsin5 sinoc 5 sinoccos5 cosocsin5 cosoc5 cosoccos5 sinocsin5 Ch45515 notes 2 sommer Complex Algebra cosoc 5 cosoccos5 sinocsin5 More Algebra in Terms of Polar inn 39 The two generic complex numbers we will use here are 2 x iy r cos6 i sin6 and w a ib p cos i sin Multiplication z 0 w xiyaib xa yb i xbya 2 pr cos6cos pr sin6sin i pr cos6sinpr sin6cos 2 pr cos6 i sin6lt Fxnnnential 39 of Complex Numbers Definition Euler s Formula ei e cos6 i sin6 NOTE This is easily derived using series expansions that will be covered later in the course Complex Number 2 x iy r cos6 i sin6 r ei 9 Complex Conjugate 2 x iy r cos6 i sin6 r e i e Modulus lzzl12 r ei er eii 812 r2 ei e4 e12 r Trigonometric Relationships ZZei0eii6 z zeie e ie cos6 72 2 and s1n6 72 i 2 i Complex Algebra Using Exponential Notation To keep things simpler we will use the following generic complex numbers 2 x iy r cos6 i sin6 r ei e w a ib p cos isin p ei Addition z wqn im6 cos1pisin1p6 em 3 2uestion How are the coefficients r p and 6 and the angles 6 I and 1p related qq 6 eW6 e419 2 r ei 9 p ei r e i 9 p e i 4 r2 pz rpei Bit 4 ei qH39 e r2 pz rpei Bit 4 67039 Bit Ch45515 notes 3 sommer Complex Algebra 2 r2 p2 rpcos6 lt i sin6 cos 6 lt i sin 6 r2 p2 rpcos6 lt i sin6 cos6 lt i sin6 r2 p2 2rp cos6 62 In other words 6 r2 p2 2rp cos6 12 Req Rezw Rez Rew 6 cos1p r cos6 p cos Imq Imzw Imz Imw 6 sin1p r sin6 p sin Definition Demoivre s Theorem If a complex number 2 x iy r cos6 i sin6 is raised to a real power 11 the expression can be rewritten z r cos6 i sin6 r cosn6 i sinn6 note Since 11 is any real number it can be fractional and or negative Example T1 12 r1 cos 6 isin 61r cos6 i sin6 Note if r1 then T1 z39 Therefore z 2 1 Z T1 cos6 2 and s1n6 2 i This can be used to quotprovequot Demoivre s theorem since ZT1n 1 n n 2 n 4 n 1 cos 6 2 n ZTH2 W2 mzn EZT I I I 211171 cosn6 cosn 26 cosn 46 1 ZT1n 1 n n 2 n 4 n 1 SUNS i 2quot i 2 Elli n 11zn n 22 Zquot Ele I I I s1nn6 s1nn 26 s1nn 46 0 l Note these were calculated using the polynomial expansion Complex roots of the number 1 Ch45515 notes 4 sommer Complex Algebra Ch45515 n0tes 5 sommer Vector Algebra Basic Concepts of Vectors For simplicity we will start by looking at two and three dimensional systems but know that the idea can be expanded infinitely Definition VECTOR A quantity that has both magnitude and directionality e g the velocity of a particle describes how fast it is going magnitude and where it is heading direction Definition SCALAR A quantity that only has a magnitude e g the speed of a particle only says how fast but does not describe a direction Definition 3 D Unit Vectors These are vectors that point in the direction of the Cartesian axes and have the magnitude of 1 x 1 The three unit vectors can be multiplied by a scalar quantity to obtain a vector pointing in the direction of the unit vector X 2 xi 6 is a scalar i is the unit vector in the x direction and X is the vector yij and zzAk A general vector R R R where R is the unit vector in the direction of R can be written in terms of its x y and z components RXyz xiy jz k Xayz R Algebra of Vectors For the purpose of examples we will use the following generic vectors a axi 1ij az k and b bxi 17ij bz k Scalar Multiplication s 0 a saxi 306lij sazAk Ch45515 notes 6 sommer Vector Algebra sabsasb Magnitude of a Vector Ial ax2 ayz azz 2 Arbitrary Unit Vector b bIbl Addition abaxbxiayby jazbz k Multiplication Several different types quotDotquot Scalar Product a 0 b Ial Ibl cos6 In terms of individual terms abzaxibay jobaz kbaxibxiby jbz k axbxiiaxbyi jaxbzi k But if we look at the individual unit vector dot products we see Ch45515 notes 7 sommer Vector Algebra i 11 cos0 1 j i Ak 11 cosTE290 0 i i Therefore a 0 b axbx ayby azbz b 0 a Note The dot product can be used as a way of finding the angle between two vectors 71 a b a b 6 cos 7 arccos 7 lal lbl lal lbl Polar Representations of Vectors This is similar to the Agrand diagrams 2 D Projections aa ia Ialcosoc iasinoc M x y J ax is said to be the quotprojection of vector a onto the x axisquot and similarly for ay Note The dot product can be considered a projection of one vector onto the other projection of b onto a Ibl cos6 aba Examples of physical quantities that are dot products projections work F 0 s F 0 ds where F is the force in the direction of the displacement s voltage 2 E 0 ds Fq ds where E is the electric field and q is a charge 3 D Projections Spherical Polar coordinates R 2 xi y Aj z Ak IRI sin6 cos i IRI sin6 sinlt Aj IRI cos6 Ak Ch45515 notes 8 SOIIIIIICF Vector Algebra quotCrossquot Vector Product a X b Ial Ibl sin6 n Where n is the unit vector quotnormalquot ie perpendicular to both vectors a and b Using this formula we can make the following conclusions ix j k and jxiz k A i and AkXAjz A kxiz j and ix kz Aj ixi jx j kx k0 Therefore in terms of unit vectors we can define the cross product as axbzaxixbay j Xbaz kxbaxixbxiby j bz k axbxixi axby x j axsziXAk axbx 0 axby k axbz j 2 clbe azby i clsz axbz Aj axby clbe Ak b X a By analogy to the dot product laXbl lallbl 6 sin 1 J arcsin laxbl lallbl Scalar Triple Product achcaXbbch x bycz bzcy y bzcx xcz z bxcy bycx 0x aybz azby Cy Clsz axbz 01 axby aybx bx cyaz czay by czax cxaz bz cxay cyax Vector Triple Product aXchbac cab Examples of physical quantities that are cross products angular velocity 0 gt v o X r where v is the linear velocity and r is the radius of rotation angular momentum L r X p r X m v m r X a X r where p is the linear momentum and m is the mass angular force torque F r X F dLdt Ch45515 notes 9 sommer Vector Algebra Ch45515 n0tes 10 sommer A work in progress CHEM 45155515 A Vector scalar or matrix variable Elements of the matrix etc are within these brackets The elements in a row may be separated by commas or by spaces the elements for different rows should be separated by 39 I Separates the limits of a range eg ab a is lower bound and b is upper where the number of elements in the range equals the integer difference between b and a or the limits and the step size eg acb a and b are the same as above but c is the interval or 39 Separates the elements of each row in a matrix OR suppresses printing when placed at the end of a statement or assignment Separates elements of a matrix or commands e g the coordinates in a plot command AB Matrix row by column multiplication A B Element by element multiplication A B Matrix division A over B AB Matrix division B over A A B Element by element division A over B A B Element by element division B over A A39 Transpose and complex conjugate of A A 39 Trargpose ofA without complex cnninoatinn Indicates that the J continues on the next line Indicates that the line is not evaluated a comment plot X Y Twodimensional plot of Y vs X must have the same number of elements in X and Y plotXYlXY2 Plots two functions of X ie Y1 and Y2 on the same graph note can use this for more than two functions by including quotXY3quot etc hold on Suppress making a plot until all quotplotquot commands are given all plots are done on the same graph regardless of type of graph so be careful polarXY Polar gra h ononX ezplot39fncx39 Graphs the function vs x without giving explicit bounds for x note the single quotes 7 may not work in a quotmquot file fplot39fncx39 x1x21 Same as above but with the bounds of x given by the quotx1quot and quotX2quot plot3XYZ Athree 139 39 graph surfXYZ Pretty much the same as quotplot3quot meshXYZ Same a quotsurfquot xlabel39string39 Assigns a label to the xaxis given by the string smiliar commands for y and z axes these commands appear on the MATLAB COMMANDS eyen or size A onesnm or ones size A zerosnm or zeros size A A B or A B or cat2AB A B or cat 1 B cat ABC A A A acosA asin A A asindA atan A COSA A n A ta A sind A tand A A cosh A B B etc MATLAB COMMANDS CHEM 45155515 same as and each other commas to A see next Ian 01 same ones 1 A zeroes 0 A Bam them into a new matrix that has all the elements of both of the matrices side side to except the elements of the matrix A on creates a a y new of matrix B rra A in the rst layer matrix B in the middle layer and a root Common Inverse arc in RADIAN units Inverse DEGREE two vectors A B vector A cross CHEM 45155515 conj z The complex conjugate of complex number 2 real z The real part of complex number 2 imag z The imaginary part of complex number 2 sumA Adds the elements of each column of matrix A if A is a row vector this returns the sum of the row elements disp string or variable Display the text written between single quotation marks string or the value of a variable or matrix eigA Displays the eigen values of matrix A VID ei9A This gives the eigen vectors in matrix V and eigen values in matrix D of matrix A Note the vectors appear as the columns in V and the values are along the diagonal of D ie this is the diagonalized version of A ode45funct range initval RungeKutta algorithm for solving first order ordinary differential equations the function describes the actual diffeq the range is for the independent variable and the initial value is for the solution This can also be used for second order diffeq s if the function treats the second derivative as the derivative of the first derivative and the first derivative as a simple differential equation The result will be a 2 by 1 matrix with y and y as the entries but the initial conditions must have two entries y0 and y 0 for counter startstepstop A programming command equivalent to a do loop while counter This is another form of the do loop where the counter is quotconditionquot checked based on some condition see next entry used when value the number of loop cycles is not predetermined if statement A decision making command the conditions are equal quotconditionquot not equal greater than gt less than lt greater value than or equal gt and less than or equal lt This is followed by the actual operation needed elseif Another decision making command that is used after a regular if command if there is another decision to be made else This is the alternative operation if the condition in the if or elseif is not met break This gets you out of a loop without completing it and without terminating the program end Closes the loop for any of the previous commands this is required to tell the program what lines are IN and what lines are not in the loop mean A Mean value of the numbers in the row matrix A median A Median value of the numbers in the row matrix A mode A The most probable value within the array MATLAB COMMANDS CHEM 45155515 std A Standard Deviation of the numbers in the row matrix A var A The variance of the numbers in row matrix A length A Counts the number of elements in the vector A cumsumA Gives the sum of the elements in a row vector bar freq data Generates a bar graph of the data in the yaXis versus the frequency in the XaXis histdatafreq Generates a histogram of the data in the XaXis versus the frequency in the yaXis NOTE If the frequency is omitted the program will bin the data into 10 evenly spaced groups histdatan Groups the data into nevenly spaced groups and generates the histogram this data may be retrieved as two vectors one with the frequencies and the other with the bin locations by typing freqbinhistdatan MATLAB COMMANDS