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File: Calculus Pdf 169257 | Multivariable Notes
lecture notes for math 417 517 multivariable calculus j dimock dept of mathematics sunyatbualo bualo ny 14260 december 4 2012 contents 1 multivariable calculus 3 1 1 vectors 3 1 ...

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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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...Lecture notes for math multivariable calculus j dimock dept of mathematics sunyatbualo bualo ny december contents vectors functions several variables limits partial derivatives the chain rule implicit function theorem i ii inverse maxima and minima dierentiation under integral sign leibniz variations vector valued other coordinate systems line integrals double triple parametrized surfaces surface area change in r gradient divergence applications more stoke s still general complex numbers denitions properties polar form special cauchy riemann equations analyticity formula higher inequalities real fourier laplace transforms we start with some a number x is positive zero or negative rational irrational denote set all label points on once pick an origin unit length are also called scalars next dene pairs elements plane pair orthogonal axes dimensional can be represented by arrows from to point triples space three especially one might em phasize that writing it bold face arrow but refrain d...

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