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MATH 200 Lecture Notes, Week 1

by: spencer.kociba

MATH 200 Lecture Notes, Week 1 Math 200

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Summary of Week 1: Introduction to 3-space, vectors, vector operations (including dot product and cross product computations). *NOTE: Section MATH 200-005 is slightly behind the other classes for ...
Multivariate Calculus
Class Notes
Math, Calculus, multivariable calculus, vectors, graphing multivariable functions in three dimension




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This 7 page Class Notes was uploaded by spencer.kociba on Sunday October 2, 2016. The Class Notes belongs to Math 200 at Drexel University taught by in Summer 2016. Since its upload, it has received 32 views. For similar materials see Multivariate Calculus in Math at Drexel University.


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Date Created: 10/02/16
MATH 200­005  Spencer Kociba Summary of Week 1: Introduction to 3­space, vectors, vector operations (including dot  product and cross product computations). *NOTE: Section MATH 200­005 is slightly behind the other classes for reasons unknown.  These are those notes for that section with Prof. Matt Ziemke. ^where you will find the syllabus, homework, homework answers and announcements Prof. Matt Ziemke Office Hours M­W, 2­3pm in Math Resource Center (Korman 249) Thurs, 2­3 in office (Korman 253) ­no required textbook, recommended one is Calculus: Early Transcendentals 10th Edition by Anton, Biven, and Davis John Wiley & Sons, 2009 ^most calc books will cover the same material in a similar order MATH 200­005  Lecture notes 09/19/2016 Spencer Kociba Multi­variable functions= functions with more than two variables Ex. f(x,y)=z; f(t)=(x,y); f(x,y,t)=z Chapter 11.1: Rectangular coordinates in 3­space, spheres, cylindrical surfaces “Euclidean 3­space’ is the graph used to portray planes, lines  and points in the 3rd dimension Ex. ***think of it like looking into the corner of a room*** “Floor” = xy­plane “Right wall” = yz­plane “Left wall” = xz­plane “Inside the room” = the first octant P= (a,b,c) = (x,y,z) **the negative axis values go “behind” or to the other side of the origin  (the axis lines extend both directions infinitely but it is easier to show the positive axes in 3­space. The negative are usually drawn using dotted or dashed lines) ­The distance, d, between the points P= (x❑ ,0y❑ ,0z❑ ) 0  and Q= (x❑ ,1y❑ ,1z❑ ) 1 is  given by d= ❑ Ex. Find d of P=(1,2,3) and Q=(0,­2,1) d= ❑ √ ❑ Sphere: a center “C=(a,b,c)” and radius “r” is the set of all points’ distances to C is r (aka all  points in 3space with a distance to C is r) d= ❑ Any point on the surface of a sphere to the center d=r 2 2 2 2 Formula:  r =(x❑ −x❑ )❑ +Cy❑ −y❑ )❑ +❑z❑ −zC )❑ ❑ C or 2 2 2 2 r =(x❑ −❑)❑ +(y❑ −b)❑ +❑z❑ −c)❑ ❑ **converse of pythagorean theorem: If  a +b =c 2  then triangle ABC must be a right  triangle MATH 200­005  Lecture notes 09/20/2016 Spencer Kociba Chapter 11.1: Rectangular coordinates in 3­space, spheres, cylindrical surfaces (cont) ­The midpoints between A= (x❑ ,0❑ ,z0 ) 0  and B= (x❑ 1y❑ ,z1 ) 1  is x❑ +0❑ 1 y❑ +0❑ 1 z❑ +0❑ 1 , , ( 2 2 2 ) Cylindrical Surface: a graph (in 3­space) of an equation with only 2 variables that is  projected all along the 3rd remaining variable’s axis Chapter 11.2: Vectors Vector: something with both direction and magnitude (most common is angle and  displacement, but these two components can be a multitude of things) Name: vector “v” or  vector AB (which is not equal to vector BA) A=initial point B=terminal point **any real number (not  ) used with vectors is  i∨ j known as a scalar  Two vectors are equal (v=w) if they have the same  length/magnitude and point in the same direction (aka POSITION  is NOT important) Zero vector: (0) the vector with no length (geometrically it  is a dot). Initial point=terminal point Vector Operations Sum: v+w=w+v  v+0=v=0+v Scalar multiplication: (scalar)*v= length*k (the direction depends on the sign of k) ^^stretch or shrink or change direction of the vector Notation:      || v || = the length or magnitude of vector v (can also put different operations of vectors in  there like || v+u || ) () indicates a point  <> indicates a vector (aka bracket notation) MATH 200­005  Lecture notes 09/21/2016 Spencer Kociba Chapter 11.2: Vectors (cont) ● 0*v=0 ● k*v=0 when v=0 ● (­1)v= ­v ; (­2)v= ­2v Subtraction:    v­w=v+(­w)     initial points are the same and you find the vector between v  and w or you change the direction by multiplying w by ­1  and take the sum  of those vectors or If vector v starts at the origin, then v= the terminal points in bracket notation= ¿v❑ ,v❑ >¿ 1 2 Vectors in Component Form For vectors v=  ¿v❑ ,1❑ >¿2  and w=  ¿w❑ ,w1 >¿ 2 , v=w ONLY if v❑ 1w❑ ∧v❑1=w❑ 2 2 k*v=  ¿kv❑ ,1v❑ >¿ 2 Component form (if initial point is NOT at the origin): v= < (x❑ 1x❑ ) 0 ,  (y❑ −2❑ ) 1 > Theorem: For any vectors u, v and w and any scalars k and l the following is true (i) u+v=v+u (ii) (u+v)+w=u+(v+w) (iii) u+0=0+u (iv) u+(­u)=0 (v) k(lu)=(kl)u (vi) k(u+v)=ku+kv (vii) (k+l)u=ku+lu (viii) 1u=u 2 v❑ ¿2 2 v❑ 1 +¿ 2 ¿∨v∨¿ =¿ ¿∨kv∨¿=(k)∗¿∨v∨¿=¿∨¿kv❑ ,kv❑ >¿∨¿ ❑ √ 1 2 ❑ Unit Vectors: Vectors that have a magnitude of 1 Ex. if v is a vector, find the unit vector u 1 u= ¿∨v∨¿ ∗v   (reciprocal of the norm of v (results in scalar) multiplied by v) Ex. v=<3,5>; ||v||=6 u= ∗¿3,5>¿< , >¿1 5 6 2 6 2 space unit vectors: i=<1,0> and j=<0,1> 3 space unit vectors: i=<1,0,0> ; j=<0,1,0> and k=<0,0,1> MATH 200­005  Lecture notes 09/22/2016 Spencer Kociba Chapter 11.2: Vectors (cont) If v is a non zero vector then  1 ∗v  = a unit vector that points in the same  ¿∨v∨¿ direction as v ^^this process is referred to as normalizing the vector v=¿∨v∨¿∗¿cosθ,sinθ>¿   θ = angle with the x axis (in 2 space)


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