TOP IN APPLIED MATH I
TOP IN APPLIED MATH I MATH 311
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This 20 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 311 at Texas A&M University taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/226051/math-311-texas-a-m-university in Mathematics (M) at Texas A&M University.
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Math 311 102 June13ID5 3mm Example image and null space Iff AX where A find the null space off and find the image off 1 0 3 x1 Null space Solve 4 6 0 x2 0 Row reduce to 0 78 16 x3 1 3W4 i 3 get 0 1 72 x2 0 Then x2t 2 where 0 o 0 x3 3 1 t is an arbitrary number The null space is a line Another interpretation The null space consists 39of all vectors that are orthogonal to every row of the matrix A June13ID5 3mm Example continued The image consists of all linear combinations of the columns of the matrix A We can find the image by column reducing 1 0 3 1 0 0 1 0 0 4 6 0 A 4 6 712 gt 4 3 0 0 78 16 0 78 16 0 74 0 The image can be described as all vectors t4f s37 4 where t and s are arbitrary numbers Another interpretation The image is the plane with equation 716x 4y 3z 0 Shortcut The dimension of the image and the dimension of the null space add up to the dimension of the domain Having shown that the dimension of the null space is 1 we can deduce that the dimension of the image is 2 so the image consists of all linear combinations of any two independent columns for instance the first two columns mm mm mm Another example Suppose x A3 and the reduced row echelon fOrm of A is 1 2 0 3 0 6 0 0 O 1 1 0 7 0 0 0 0 1 4 0 Say as much as possible about the 0 0 0 0 0 0 0 0 0 0 0 0 0 0 domain range null space and image off Solution The domain is R7 and the range is 1R5 Forthe null space we can choose x2 x4 x5 and x7 arbitrarily and then x1 X3 and x5 the leading variab es will be determined The null space is 4 dimensional and consists of the vectors x272 10 oooo X4730711000 x5 76 o 75 o 41 1 o mm 0 e7 0 0 01 The image is 3dimensional a linear combinations of columns 1 3 and 5 of the original matrix not the rowreduced matrix mm mm m A basis for a vector space is a set of vectors satisfying any one of the following equivalent descriptions 1 a maximal linearly independent set 2 a minimal spanning set 3 a set such that every vector can be represented in a unique way as a linear combination of elements of the basis Examples The standard basis forJR2 is the pair of vectors 10 and 01 Another basis forJR2 is the pair of vectors 1 1 and 171 The standard basis for the space 732 of polynomials of degree less than or equal to 2 is 1 x x2 Another basis for 732 is the set of socalled Legendre polynomials 1 x and 3x2 71 The dimension of a vector space is the number of elements in a basis All bases have the same number of elements June 3 m5 slam Example bases The set of all linear combinations of sinx and cosx is a twodimensional vector space a subspace of the vector space of differentiable functions For the basis sinx cosx what matrix represents the linear operator of differentiation Solution Since the derivative of the first basis element equals the second and the derivative of the second basis element equals the negative of the first the matrix is a Continuation Euler s formula says 2 cosx isinx where i is the complex number whose square is 71 What matrix represents the operator of differentiation with respect to 39 4 39 o the alternate basis e e 11 Answer 71gt A linear transformation is easiest to understand if a basis is used in which the representing matrix is diagonal June 3 m5 sliden Math 311102 mm ma slider Eigenvectors and eienvalues A nonzero vector 393 is an eigenvector of a matrix A if A6 is a scalar multiple of 3 1 Example IfA2 1 1 1 and 2 1 1 0 then A6 2 2 2 26 0 Thus is an eigenvector ofA with eigenvalue equal to 2 How can you find eigenvectors The strategy is to find the eigenvalues first mu ma slide Finding eienvalues Given a matrix A the goal is to find a nonzero vector 17 and a scalar A such that Ad Ad An equivalent equation is A MW 2 0 where I is the identity matrix There can be such a nonzero vector only if the matrix A AI fails to be invertible The equation detA AI 2 0 determines the eigenvalues it is called the characteristic equation of the matrix A 413 5 Exam le Find the el envalues of p g 30 12 Solution Solve det IS A 5 oor 30 12 A 13 A12 1502Oori2i 62Oor A 3i 2 0 The eigenvalues are 2 and 3 mm mm was Findin eienvectors Continuing the example the matrix A 2 43 5 has 30 12 eigenvalues 2 and 3 Find corresponding eigenvectors Solution The eigenvector 17 corresponding to eigenvalue 2 711 satisfies A 21W 0 or 45 2 0 Row v2 5 3o 10 reducing gives 3 0 so 2 2 ilgtgt2l Similarly solving A 31w 2 0 or 155 2 0 gives as an eigenvectorwith eigenvalue 3 mm mm slim Chane of basis 13 5 30 12 represents a linear transformation with respect to the standard basis 1 What matrix represents the same transformation with respect to the basis of eigenvectors The matrix A 2 Answer The diagonal matrix D 7 whose diagonal entries are the eigenvalues The matrix 11 2 1 1 whose columns are the eigenvectors translates from eigenvector coordinates to standar coordinates The matrix 11 1 2 j j translates from standard coordinates to eigenvector coordinates Then A 2 UDU l and 11211411 2 D Check We can diagonal26 a matrix by using a basis of eigenvectors mu xn slam Math 311 102 Jim m5 slider Reminder The comprehensive final exam is 100 300PM Tuesday July 5 in this room Please bring paper or a bluebo ok to the exam The exam covers everything on the syllabus There are 10 questions on the exam rum m5 slide Review problems vector calculus I Evaluate the line integral C F d When Fxyz zxzxy and C is the curve described parametrically by gt cost1 sintt 0 g t g 271 I Evaluate the surface integral f5 zzda when S is the surface described by z v 1 7x2 iyz z 2 0 I Evaluate the volume integral fffB x2 dxdydz when B is the ball described by x2 y2 22 g 1 I Evaluate the flux integral ffs F Fido when Fxyz y2z2x2 and S is the open surface described parametrically by guv uv17u2 ivz u v2 lt1 with normal vector oriented upward m4 mm mm Review problems vector calculus I Considerthe unit cube in R3 determined by the three vectors 100 010 and 001 Orient the surface with the outwardpointing normal vector Let S be the union of the five faces of the cube on which z gt 0 Evaluate the integral ffs cuxlF Fido when Fxyz y2z2x2 I Find a point abc on the curve t 6832 and a point def on the surface guv um u2v2 such that the tangent line to the curve at 1117c does not intersect the tangent plane to the surface at def I Construct an invertible coordinate transformation x g u in some region ofJR2 such that the area y 390 element transforms via 171x lily 2 du 171 m4 m5 Elm Review problems linear alebra 734 716 36 I If A 718 76 18 find a 3 x 2 matrix B such that 747 721 49 ABB 2 L0 0 3 lSuppose l 1 2 16391 and172 2 Find a matrix B such that if 111 11 bli l 172 then 111 B 171 I 3912 2 I True or false If V1 and V2 are two subspaces of R3 both of dimension 2 then there exists a lineartransformation f R3 A R3 such thatfV1 V2 Juiy1x 5 3mm Review problems linear alebra I Construct a nonstandard inner product on R2 for which the vectors 10 and 11 become orthogonal I Give an example of a lineartransformation f R3 a R2 such that the vectors 10 and 11 form a basis forthe image and the vector 24 2 is a basis for the null space I In the vector space 73 of polynomials express x as a linear combination of p1x 1 xx2p2x12x3x2 and p3x 27 x4x2 I The trace of a square matrix is the sum of the elements on the main diagonal for example iracelt 1 gt 159 15 On the vector space of3 X 3 matrices is the trace a linear function Juiy1x 5 3mm Math 311102 Maw Inns gum Two examples of motivating problems 1 Solve a system of linear equations such as 2x1 3x 7 9x1 7 5x 4 The mathematics involved with thousands of variables underlies the inputoutput method in economics for which Wassily Leontief won the 1973 Nobel Prize 2 Explain electromagnetism and the propagation of light VE47Ip VB 0 Maxwell s equations V X E 71 C at 13E 471 V x B 7 E TJ Mayamnns 5mm Today s topic vectors Chapter 1 A vector is both a geometric object and an algebraic object As a geometric object a vector has a length and a direction Example Find the length of the vector joining one corner of a unit cube to the opposite corner and find the angle the vector makes with a side Solution The vector 6 v1vzv3 may be written as 111 or 111 orfor 1 52 63 By the Pythagorean theorem the length 6 is x1212 12 V5 A unit vector pointing in the same direction is 73 The vector makes equal angles with the coordinate axes namely arccos m 54736 m 0955 radians Mam Inns gum Example collision course Problem You are in a sailboat moving at a constant speed on a fixed heading A sailboat on a different heading is coming closer How can you tell if you are on a collision course Solution Line up the other boat with a point on the distant shore Ifthat point doesn t move you are on a collision course Mathematical explanation Your motion along a straight line can be described parametrically as t Where it is your position at time t 0 and i is yourvelocity vector The position of the other boat may be written similarly as 17 wt The vector pointing from your boat to the other one is the difference vector 17M E 7 31 73 7 t If this vector is 0 for some t then the vectors 37 it and 7D 76 are parallel Then 58 points in the same direction for every t The parallel lines in the direction Ea meet at infinity that is at a point on the distant shore Maw mg ml The algebra of vectors Vectors can be added and can be multiplied by scalars and the operations satisfy the associative commutative and distributive laws Example Can the vector 6071947219 be written as a linear combination of the vectors 123 and 987 Solution Only if we are lucky because the vector equation x1 23 y98 7 607 194 7219 translates to a system of three simultaneous equations in two unknowns Dot product By the Pythagorean theorem the square of the length of a vector 3 x1x2x3 is ifiz x x 33 This suggests introducing a scalar product of vectors 3 X1 x2 X3 and y ylmzlys Via 339 V lel Xzyz 3 3y3 Then 13 ifiz By the law of cosines i327in RF We 2m iyi cos9 which simplifies to 25H 39 ifii i cos9gt z where 9 is the angle between the vectors Z and y There is nothing special about dimension 3 analogous formulas apply to the dot product of vectors in any dimension Mammals 5mm x 9y 607 2x By 194 3x 7y 7219 We are lucky for x 7311 and y 102 works in all three equations Maw was Ms Projection The projection of a vector i onto a vector 73 is atam where u3 Example A bicycle travels 3 kilometers northeast against an easterly wind that makes a resistive force of 20 newtons How much work is done by the cyclist Solution Only the component of the force F in the direction of the displacement 11 does work so the work equals F for 1 N 20 x 3000 x W N 42426Joules Mam Inns gum Cross product The vectorproduct of two threedimensional vectors a 141142143 and a v1v 3 isavector x77 perpendicularto both a and dwith length equal to i i iii 51119 where 9 is the angle between 11 and 6 and with direction determined by the right hand rule In particular x f E ifx Ix F 7 gtlt and F gtlt I f efx F The cross product is anticommutative so 6 x 17 0 for every vector 5 Example 2 3 X 4f 5E 86x 106x F 15fgtlt E 1511 1of 8E The cross product is special to threedimensional vectors Maw 2m max Determinant form for cross product Geometry and the cross product The length of the cross product l6 gtlt wl equals the area of the parallelogram determined by the vec tors 5 and 7D 5 The absolute value of the scalar tripe product la 6 gtlt ml equals the volume of the paral w v2 leepiped determined by the vectors H 6 and 7D vs s 4 14233 i 14332 L 14133 i 14331 3914le i 11231 k Example revisited The scalar triple product can be written as a determinant f E 1 142 143 23Dx4f5 2 3 0 WWW v1 v2 v3 0 4 5 W1 2 7HS 3 o 7 2 a 2 3 s s s a Wewill usethisdeterminantlater in Jacobi stheorem about 4 5 17 o 5 o 4 k151 108k39 change ofvariablesin multiple integrals Chapter7 Maw Inns gum May t z s 5mm Math 31 1102 Change of variables in integrals via the change ofvariable u 1x2 What happens in higher dim ensions To see what happens the place to start is with a linear transformation in n m m l Change of variablequot means coordinate transform ationquot Remi The nder second examination is Thursday June 23 Linear functions Exampl tices at e Let S be the square ofarea 1 with ver 00 1 0 11 and 01 in 47 space Let y 7 so the image of S in any space is a parallelo ram P The rea of equals 77 2739 gtlt 27l the length of the cross product which equals 4 Thus P dandy 4 54dudv trans Sum abso form ation 7 The factor of4 is the determinant ofthe coordinate 1 2 mary A linear transformation magni es area by the lute value of the determinant Jacobi s theorem For any invertible coordinate transformation T not necessarily linear from a region R in W space to xy space limeow My HRHTWD idem1W dudv and similarly for transformations in R3 Example polar coordinates The Jacobian matrix of the coordinate transformation x rc se equals 1 151119 cf se 715mg the determinant is 1 so st 1cosQ ff x2y2dxdy W 11119 xzyzg1 006221 7 7 S 0112111 02 119 2713 mm mm slam Curvilinear coordinates The notation for cylindrical coordinates in R3 is 192 where 19 are polar coordinates in the xy plane The volume element 11x11de transforms to 1111 119112 The notation for spherical coordinates in R3 depends on the age of the book and on the subject mathematics or physics The distance from a pointto the origin is denoted by r or p In modern mathematics books 9 denotes the same angle as in cylindrical coordinates and 4 denotes the angle measured down from the z axis Older mathematics books and many physics and engineering books reverse the meanings of 9 and 4 The volume element 11x 11y 112 transforms to 72 sinangle down from the z axis 1111191141 mm mm sliden Math 311 102 June17ID5 slider Tanent approximation For functions of one variable the tangent approximation formula says fx m u f ax 71 Example If fx 5 and a 1 the formula says 9 s 1 ltx 71 The multi variable approximation formula is a a My erltmbgt ltwgtltxeagt ltaibgtltyebgt Example lffxy sinxy 4y and 1117 00 then sinxy 4y m sin0 cos0x 70 2cos0y7 0 2y June17ID5 slide Approximation of vector functions XZJr 2 Example To approximate fxy x3 siny near the V point 1117 use the previous formula in each row to get 211x e 10 217W 7 b rltxygtefltabgt sa2ltxeagtcosltbgtltyebgt ebxitzaebyib 2 ltgt b f1 b 3112 cosb Eb Heb We can think of the derivative of a vector function as being a matrix mm mm mm The derivative matrix The preceding example shows that the derivative of a function f1 Xi y may f2ltxiygt f3 39y should be viewed as the matrix of partial derivatives 8 a 1 Th 2 a a a li aria The rows of the matrix are the gradients of the component functions The matrix is often called the Jacobian matrix mm mm Elm Exercise SW3 911W Z x 1 siny 7 222 7 cosx y 112 1 Find the derivative matrix off at 000 NW Z 2 Show that the matrix is orthogonal the inverse equals the transpose 3 The matrix is a rotation matrix Find the axis of rotation June17 ID5 slam 1 The Jacobian matrix equals f C0596 1 secz V g cosltygt 7 cosx x sinx z y 000 1 g A 39 2 25 5 g L5 a 254 g 5 5 5 0 2 The columns of the matrix are orthonormal 3 3 The vector A is an eigenvector with eigenvalue 1 0 June17 ID5 sliden Math 311 102 Harold P Boas boastamu edu Parametric curves in space Example The vectorfunctionft costsintt represents a curve in R3 a helix Find an equation for the line tangent to the curve at the point where t 712 Solution The derivative f t gives the velocity vector tangent vector isintcost 1 lr z 7101 The point on the curve corresponding to t 712 is 01n2 A parametric equation for the tangent line in terms of parameters is 01n2s7101 Continuation What is the arc length of the curve from t 0 to t 27 Solution To get the arc length integrate the speed length of the velocity vector fez lf tl dt 02 sin2 1 coszt1dt 02 dt 2n Parametric surfaces Directional derivatives Example As u and v vary from 0 to 271 the vector function Exampe If fxyz x2y ng find the rate of change ofr fer 5 251D C05vr5 251D Sinvrzcosu sweeps in the direction of the unit vector 7 g at the point 210 out a surface in R3 a torus Find an equation forthe plane tangent to the surface at the point where u 716 and v 712 Solution Each of the three partial derivatgfes ofa contzrifbutes dth t fh lth 27737 i Solution The point on the surface is 0 6 The partial an 2 e a e c nge ejua S e sumlgnr 731 732 210 derivative is a vectortangent to the surface GOXW e 7 7x 73ml 210 739 2 cosu cosv 2 cosu sinv 72 sinu u7I5 0 71 Notation The gradient off written Vf means the vector v7r2 a a a a 2 75 zslnu51nv 5 251nucosv 0gt 5 76 0 0 Vf a The length lVfl represents the largest rate of change The cross product of the two tangent vectors gives a vector 0f f in any direCtion normal to the surface 066 Just as good a normal is the scalar multiple 01 6 The tangent plane has equation 0X 01y76 27 Oory z9 Mus m mm mm 12 mm m We mm 112 Math 311 102 m2 m5 slider Overview Yesterday systems of equations and row reduction 2 1 x 12 Forexample the system 4 3 711 y 1 3 5 71 728 z 717 has the same solutions as the reduced system 1 0 75 x 72 0 1 3 y 7 0 0 0 z 0 Today matrix algebra sums products inverses For example find a matrix M such that the matrix product 1 2 1 1 0 75 M 4 3 711 0 1 3 5 71 728 0 0 0 m2 m5 slide Componentwise operations Sums and differences of matrices are computed componentwise Examle10207127 9 18 p39304o 34 273639 Multiplication of a matrix by a scalar is computed componentwise 1 2 10 20 Exam le 10 p 3 4 30 40gt The product of a matrix with a matrix however is not computed componentwise m2 mm mm Matrix multiplication The product of a matrix with a column vector is a new vector obtained by taking dot products with the rows of the matrix 1 2 3 1 321 4 5 6 10 654 7 8 9 100 987 The product of a matrix with another matrix is computed similarly by letting the first matrix act on each column of the 0 2 1 5 201 1005 second matrix 0 3 4 10 50 430 2150 5 6 0 100 500 65 325 The product of matrices makes sense only if the rows of the first matrix have the same number of entries as the columns of the second matrix m2 m5 Elm Multiplication is not commutative Example The order of multiplication matters AB y BA On the other hand matrix multiplication is associative ABC ABC Notice in the above example that the product of two nonzero matrices can be the zero matrix In particular only certain matrices have multiplicative inverses m2 mm mm Identity and inverses For 3 x 3 matrices the identity matrix I property that IA A and AI A for every 3 X 3 matrix A and similarly for square matrices of other sizes Two matrices A and B are inverses if AB I and BA I For 2 gtlt 2 matrices there is a simple rule forthe inverse of a matrix 11 b 17 1 d 7b c d fadibc 7c 1139 This inverse exists if and only if the determinant 11117 be 7 0 m2 mm mm Computin inverses in eneral Method write the matrix next to an identity matrix and do row operations on both matrices simultaneously When the initial matrix has been turned into the identity matrix the identity matrix will have been turned into the inverse matrix Example 1071 100 2 1 0 0 1 0 rowreduceq 72 0 1 0 0 1 1 0 71 1 0 0 1 0 0 71 0 1 0 1 2 72 1 0 gt 0 1 0 2 1 2 71 2 0 1 0 0 1 72 0 71 71 1 71 o 71 Conclusion 2 1 0 2 1 2 72 0 1 72 0 71 m2 mm mm Math 311 102 ma m5 slider About the exam The first examination is tomorrow Thursday June 9 Please bring paper or a bluebook to the exam The exam covers everything on the syllabus to date There are 10 questions on the exam The types of questions include calculation application and in terpreta lion ma m5 slide Examples of calculation Calculation with vectors length dot product cross product projection Calculation with matrices sums and products reduced matrices inverse matrices determinants solving a system of equations mg mm mm Examples of application With vectors angles work area of a parallelogram or of a triangle volume of a parallelepiped equations of lines and planes intersections of lines and planes With matrices writing a vector as a linear combination of other vectors testing linear independence ofvectors solving a system by Cramer s rule m3 m5 slat Examples of interpretation Is a certain set a vector space Is a certain subset a subspace Is a certain function linear onetoone invertible Does a system of equations have a unique solution infinitely many solutions no solutions What does a certain linear transformation represent geometrically ma mm mm Math 311 102 Onevariable chain rule Example You learned in your first calculus course that 1n5111x c05x The general rule for differentiating a composite function f0 g is 11m f gxgtgtg xgt M 7 dy d Alternate notation lfy fu and u gx then a 7 Example If you forgot the formula for 1 5111 1x how could you work it out Solution Put y 5111 1x which implies that 5111y x We want to find and the chain rule implies that this equals 3 F5165 Since c05y 17 51112y 1 7x2 it follows that 5111T1x 11 Z 7x mm mm slide Linear functions x 1 2 3 x Example lff y 4 5 6 y what is the derivative 7 8 9 z Z matrix Jacobian matrix of f Answer The derivative matrix has rows equal to the gradients of the component functions of f so the Jacobian matrix is 1 2 3 4 5 6 7 8 9 In other words a Iinearfunction is its own linear approximation You compose linearfunctions by multiplying their matrices Therefore the derivative matrix of a multivariable composite function equals the product of the derivative matrices Junem mm mm Example multivariable chain rule x x y x 1 rc059 7 2 7 7 Mfg 7 xxy and ltygt 7glt9gt 7 lt1Sin9gt polar coordinates find the derivative of the composite function f0 3 Solution The derivative matrix off with respect to x and y 1 equals 2x 1 21c059 1 and the derivative y x r 51119 1 c059 c059 71 51119 81119 r COS9gt the product equals matrix ofg equals lt c05951119 7151119 rc059 21c0529 51119 7272 c059 511191c059 21 51119 c059 12 51139 12 c052 Junem mm m u 81 89 av av 81 89 aw aw Br 89 For exam ple Example continued u Alternate notation If v 74 x 7 1cos9 y 7 lt151n9gtY then Bu Bu 3 a 7 av av T a 27 3 3 Bx By 978x89 x2y and W E 81 as a a 87 89 y ByBQ39 mm mm slam Coordinate transformations Question Is the polar coordinate transformation x 959 invertible y 73119 Answer You can solve explicitly for1 xx2y2 and 9 tan 1 cot 1 i but there are two problems There is a global problem that 9 is ambiguous by addition of multiples of 271 There is a local problem that 9 is not defined when x y 0 The local problem is addressed by the Inverse function theorem A transformation is locally invertible at points where the Jacobian matrix is invertible Example Since det 1 the theorem confirms that the polar coordinate transformation is locally invertible when 1 i 0 mm mm sliden
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