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