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Lecture notes for Math 417-517 Multivariable Calculus J. Dimock Dept. of Mathematics SUNYatBuffalo Buffalo, NY 14260 December 4, 2012 Contents 1 multivariable calculus 3 1.1 vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 the chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 implicit function theorem -I . . . . . . . . . . . . . . . . . . . . . . . . 14 1.8 implicit function theorem -II . . . . . . . . . . . . . . . . . . . . . . . . 18 1.9 inverse functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.10 inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.11 maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.12 differentiation under the integral sign . . . . . . . . . . . . . . . . . . . 29 1.13 Leibniz’ rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.14 calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 vector calculus 36 2.1 vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 other coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.4 line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 double integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.6 triple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.7 parametrized surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1 2.8 surface area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.9 surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.10 change of variables in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.11 change of variables in R3 . . . . . . . . . . . . . . . . . . . . . . . . . 67 3 2.12 derivatives in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.13 gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.14 divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.15 applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.16 more line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.17 Stoke’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.18 still more line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.19 more applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.20 general coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . 99 3 complex variables 106 3.1 complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2 definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.3 polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.4 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.5 special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.6 derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.7 Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.8 analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.9 complex line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.10 properties of line integrals . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.11 Cauchy’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 3.12 Cauchy integral formula . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.13 higher derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.14 Cauchy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.15 real integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.16 Fourier and Laplace transforms . . . . . . . . . . . . . . . . . . . . . . 137 2 1 multivariable calculus 1.1 vectors We start with some definitions. A real number x is positive, zero, or negative and is rational or irrational. We denote R=set of all real numbers x (1) The real numbers label the points on a line once we pick an origin and a unit of length. Real numbers are also called scalars Next define 2 R =all pairs of real numbers x = (x ,x ) (2) 1 2 The elements of R2 label points in the plane once we pick an origin and a pair of orthogonal axes. Elements of R2 are also called (2-dimensional) vectors and can be represented by arrows from the origin to the point represented. Next define 3 R =all triples of real numbers x = (x ,x ,x ) (3) 1 2 3 3 The elements of R label points in space once we pick an origin and three orthogonal 3 3 axes. Elements of R are (3-dimensional) vectors. Especially for R one might em- phasize that x is a vector by writing it in bold face x = (x ,x ,x ) or with an arrow 1 2 3 ~x = (x ,x ,x ) but we refrain from doing this for the time being. 1 2 3 Generalizing still further we define n R =all n-tuples of real numbers x = (x ,x ,...,x ) (4) 1 2 n n The elements of R are the points in n-dimensional space and are also called (n- dimensional) vectors Given a vector x = (x ,...,x ) in Rn and a scalar α ∈ R the product is the vector 1 n αx=(αx ,...,αx ) (5) 1 n Another vector y = (y ,...,y ) can to added to x to give a vector 1 n x+y=(x +y ,...,x +y ) (6) 1 1 n n n Because elements of R can be multiplied by a scalar and added it is called a vector space. We can also subtract vectors defining x − y = x + (−1)y and then x−y=(x −y ,...,x −y ) (7) 1 1 n n For two or three dimensional vectors these operations have a geometric interpreta- tion. αx is a vector in the same direction as x (opposite direction if α < 0) with length 3 Figure 1: vector operations increased by |α|. The vector x + y can be found by completing a parallelogram with sides x,y and taking the diagonal, or by putting the tail of y on the head of x and drawing the arrow from the tail of x to the head of y. The vector x − y is found by drawing x+(−1)y. Alternatively if the tail of x−y put a the head of y then the arrow goes from the head of y to the head of x. See figure 1. Avector x = (x ,...,x ) has a length which is 1 n q 2 2 |x| = length of x = x1 +···+xn (8) Since x−y goes from the point y to the point x, the length of this vector is the distance between the points: p 2 2 |x −y| = distance between x and y = (x −y ) +···+(x −y ) (9) 1 1 n n Onecanalso form the dot product of vectors x,y in Rn. The result is a scalar given by x·y =x y +x y +···+x y (10) 1 1 2 2 n n Then we have x·x=|x|2 (11) 4
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