Lecture notes for Math 417-517 Multivariable Calculus J. Dimock Dept. of Mathematics SUNY at Buffalo 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 2.12 derivatives in R3 .............................. 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 = (x1; x2) (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 = (x1; x2; x3) (3) The elements of R3 label points in space once we pick an origin and three orthogonal axes. Elements of R3 are (3-dimensional) vectors. Especially for R3 one might em- phasize that x is a vector by writing it in bold face x = (x1; x2; x3) or with an arrow ~x = (x1; x2; x3) but we refrain from doing this for the time being. Generalizing still further we define n R = all n-tuples of real numbers x = (x1; x2; : : : ; xn) (4) The elements of Rn are the points in n-dimensional space and are also called (n- dimensional) vectors n Given a vector x = (x1; : : : ; xn) in R and a scalar α 2 R the product is the vector αx = (αx1; : : : ; αxn) (5) Another vector y = (y1; : : : ; yn) can to added to x to give a vector x + y = (x1 + y1; : : : ; xn + yn) (6) Because elements of Rn 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 = (x1 − y1; : : : ; xn − yn) (7) 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 jαj. 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. A vector x = (x1; : : : ; xn) has a length which is q 2 2 jxj = 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 jx − yj = distance between x and y = (x1 − y1) + ··· + (xn − yn) (9) One can also form the dot product of vectors x; y in Rn. The result is a scalar given by x · y = x1y1 + x2y2 + ··· + xnyn (10) Then we have x · x = jxj2 (11) 4 1.2 functions of several variables We are interested in functions f from Rn to Rm (or more generally from a subset D ⊂ Rn to Rm called the domain of the function). A function f assigns to each x 2 Rn a point y 2 Rm and we write y = f(x) (12) The set of all such points y is the range of the function. n Each component of y = (y1; : : : ; ym) is real-valued function of x 2 R written yi = fi(x) and the function can also be written as the collection of n functions y1 = f1(x); ··· ; ym = fm(x) (13) If we also write out the components of x = (x1; : : : ; xn), then are function can be written as m functions of n variables each: y1 =f1(x1; : : : ; xn) y =f (x ; : : : ; x ) 2 2 1 n (14) ::: ym =fm(x1; : : : ; xn) The graph of the function is all pairs (x; y) with y = f(x). It is a subset of Rn+m. special cases: 1. n = 1; m = 2 (or m = 3). The function has the form y1 = f1(x) y2 = f2(x) (15) In this case the range of the function is a curve in R2. 2. n = 2; m = 1. Then function has the form y = f(x1; x2) (16) The graph of the function is a surface in R3. 3. n = 2; m = 3. The function has the form y1 =f1(x1; x2) y2 =f2(x1; x2) (17) y3 =f3(x1; x2) The range of the function is a surface in R3. 4. n = 3; m = 3. The function has the form y1 =f1(x1; x2; x3) y2 =f2(x1; x2; x3) (18) y3 =f3(x1; x2:x3) The function assigns a vector to each point in space and is called a vector field. 5 1.3 limits Consider a function y = f(x) from Rn to Rm (or possibly a subset of Rn). Let x0 = 0 0 n 0 0 0 m (x1; : : : xn) be a point in R and let y = (y1; : : : ; ym) be a point in R . We say that y0 is the limit of f as x goes to x0, written lim f(x) = y0 (19) x!x0 if for every > 0 there exists a δ > 0 so that if jx − x0j < δ then jf(x) − y0j < . The function is continuous at x0 if lim f(x) = f(x0) (20) x!x0 The function is continuous if it is continuous at every point in its domain. If f; g are continuous at x0 then so are f ± g. If f; g are scalars (i.e. if m = 1) then the the product fg is defined and continuous at x0. If f; g are scalars and g(x0) 6= 0 then f=g is defined near x0 and and continuous at x0. 1.4 partial derivatives At first suppose f is a function from R2 to R written z = f(x; y) (21) We define the partial derivative of f with respect to x at (x0; y0) to be f(x0 + h; y0) − f(x0; y0) fx(x0; y0) = lim (22) h!0 h if the limit exists. It is the same as the ordinary derivative with y fixed at y0, i.e d f(x; y ) (23) dx 0 x=x0 We also define the partial derivative of f with respect to y at (x0; y0) to be f(x0; y0 + h) − f(x0; y0) fy(x0; y0) = lim (24) h!0 h if the limit exists. It is the same as the ordinary derivative with x fixed at x0, i.e d f(x ; y) (25) dy 0 y=y0 6 We also use the notation @z @f f = or or z x @x @x x (26) @z @f f = or or z y @y @y y If we let (x0; y0) vary the partial derivatives are also functions and we can take second partial derivatives like @ @z @2z (f ) ≡ f also written = (27) x x xx @x @x @x2 The four second partial derivatives are @2z @2z f = f = xx @x2 xy @y@x (28) @2z @2z f = f = yx @x@y yy @y2 Usually fxy = fyx for we have Theorem 1 If fx; fy; fxy; fyx exist and are continuous near (x0; y0) (i.e in a little disc centered on (x0; y0) ) then fxy(x0; y0) = fyx(x0; y0) (29) Example: Consider f(x; y) = 3x2y + 4xy3. Then 3 2 2 fx = 6xy + 4y fy = 3x + 12xy 2 2 (30) fxy = 6x + 12y fyx = 6x + 12y n We also have partial derivatives for a function f from R to R written y = f(x1; : : : xn).