MULTIVARIABLE CALCULUS

MULTIVARIABLE CALCULUS JOHN QUIGG Contents 13.1 Three-Dimensional Coordinate Systems 2 13.2 Vectors 3 13.3 The Dot Product 5 13.4 The Cross Product 6 13.5 Equations of Lines and Planes 7 13.6 Cylinders and Quadric Surfaces 8 14.1 Vector Functions and Space Curves 9 14.2 Derivatives and Integrals of Vector Functions 10 14.3 Arc Length 11 14.4 Motion in Space: Velocity and Acceleration 12 15.1 Functions of Several Variables 13 15.2 Limits and continuity 14 15.3 Partial Derivatives 15 15.4 Tangent Planes and Linear Approximations 17 15.5 The Chain Rule 18 15.6 Directional Derivatives and the Gradient Vector 19 15.7 Maximum and Minimum Values 20 15.8 Lagrange Multipliers 21 16.1 Double Integrals over Rectangles 22 16.2 Iterated Integrals 23 16.3 Double Integrals over General Regions 24 16.4 Double Integrals in Polar Coordinates 25 16.5 Applications of Double Integrals 26 16.7 Triple Integrals 27 13.7 Cylindrical and Spherical Coordinates 29 16.8 Integrals in Cylindrical and Spherical Coordinates 30 17.1 Vector Fields 31 17.2 Line Integrals 32 17.3 The Fundamental Theorem for Line Integrals 33 17.5 Curl and Divergence 34 17.6 Parametric Surfaces 35 17.7 Surface Integrals 36 17.8 Stokes Theorem 37 17.9 Divergence Theorem 38 Date: October 23, 2003. 1

2 JOHN QUIGG 13.1 Three-Dimensional Coordinate Systems 3-d space: R 3 := {(x, y, z) : x, y, z R} origin: O = (0, 0, 0) x-axis: {(x, y, z) R 3 : y = z = 0} y-axis: {(x, y, z) R 3 : x = z = 0} z-axis: {(x, y, z) R 3 : x = y = 0} xy-plane: {(x, y, z) R 3 : z = 0} xz-plane: {(x, y, z) R 3 : y = 0} yz-plane: {(x, y, z) R 3 : x = 0} distance between P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ): P 1 P 2 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 (z 2 z 1 ) 2 sphere centered at C(h, k, l) with radius r: (x h) 2 + (y k) 2 + (z l) 2 = r 2

MULTIVARIABLE CALCULUS 3 13.2 Vectors 2-d vector: a = a 1, a 2 a 1, a 2 R components representation as directed line segment: a = P Q, initial point P = (x, y), final point Q = (x + a 1, y + a 2 ) position vector of P : r = OP where O = (0, 0) here 3-d vector: a = a 1, a 2, a 3 obvious analogues of 2-d vector concepts magnitude: a = a 2 1 + a2 2 + a2 3 also length, also written a in textbook zero vector: 0 = 0, 0, 0 vector addition: a 1, a 2, a 3 + b 1, b 2, b 3 = a 1 + b 1, a 2 + b 2, a 3 + b 3 scalar multiplication: c a 1, a 2, a 3 = ca 1, ca 2, ca 3 c R scalar = real number properties: ca = c a ca has same direction as a if c > 0 ca has opposite direction from a if c < 0 parallel vectors: negative vector: one is scalar multiple of the other a = ( 1)a vector subtraction: a b = a + ( b) also difference

4 JOHN QUIGG properties: a + b = b + a a + (b + c) = (a + b) + c a + 0 = a a a = 0 c(a + b) = ca + cb c R (c + d)a = ca + da (cd)a = c(da) 1a = a standard basis vectors: i = 1, 0, 0 j = 0, 1, 0 k = 0, 0, 1 a 1, a 2, a 3 = a 1 i + a 2 j + a 3 k 2-d: i = 1, 0 j = 0, 1 a 1, a 2 = a 1 i + a 2 j unit vector: a = 1 unit vector in same direction as a: a a

MULTIVARIABLE CALCULUS 5 13.3 The Dot Product dot product: a 1, a 2, a 3 b 1, b 2, b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3 properties: a a = a 2 a b = b a a (b + c) = a b + a c (ca) b = c(a b) = a (cb) 0 a = 0 a b = a b cos θ θ = angle between a, b orthogonal vectors: a b = 0 also perpendicular scalar projection of b on a: comp a b = b cos θ b θ C D a vector projection of b on a: properties: proj a b = CD comp a b = a b ( a ) a b proj a b = a = ( comp a a a b ) a a work done by force F in moving particle from P to Q: W = F P Q = ( comp P Q F ) P Q

6 JOHN QUIGG 13.4 The Cross Product cross product: a 1, a 2, a 3 b 1, b 2, b 3 = a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 properties: a b = b a a (b c) (a b) c) a (b + c) = a b + a c (a + b) c = a c + b c (ca) b = c(a b) = a (cb) c R a b perpendicular to both a and b a, b, a b form a right-handed system of vectors a b = a b sin θ = area of parallelogram with adjacent edges a, b θ = angle between a, b a parallel to b a b = 0 triple scalar product: a (b c) = (a b) c properties: a (b c) = volume of parallelepiped with adjacent edges a, b, c a, b, c coplanar a (b c) = 0

line through P 0 parallel to vector v: MULTIVARIABLE CALCULUS 7 13.5 Equations of Lines and Planes vector equation: r = r 0 + tv t R parameter P 0 P = r r 0 parallel to v parametric equations: x = x 0 + at, y = y 0 + bt, z = z 0 + ct P = (x, y, z) P 0 = (x 0, y 0, z 0 ) v = a, b, c x x 0 symmetric equations: = y y 0 a b skew lines: do not intersect, not parallel = z z 0 c (= t) distance from point P to line through P 0 parallel to v: θ = angle between P 0 P, v plane through P 0 perpendicular to vector n: vector equation: n (r r 0 ) = 0 scalar equation: parallel planes: a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 ax + by + cz = d (= ax 0 + by 0 + cz 0 ) linear equation in x, y, z normal vectors parallel distance from point P to plane through P 0 normal to n: θ = angle between P 0 P, n P 0 P sin θ = P0 P v v P 0 P = r r 0 perpendicular to n (P = (x, y, z), P 0 = (x 0, y 0, z 0 ), n = a, b, c ) P 0 P cos θ = comp n P0 P = P0 P n n

8 JOHN QUIGG 13.6 Cylinders and Quadric Surfaces quadric surface: graph of quadratic equation in x, y, z Ax 2 + By 2 + Cz 2 + Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0 parabolic cylinder: y = x 2 parallel to z-axis horizontal traces: parabolas opening in positive y-direction elliptic cylinder: x 2 + 2y 2 = 1 parallel to z-axis horizontal traces: ellipses hyperbolic cylinder: x 2 2y 2 = 1 parallel to z-axis horizontal traces: hyperbolas opening in x-direction elliptic paraboloid: z = x 2 + 2y 2 horizontal traces: ellipses vertical traces: parabolas opening up bowl-shaped hyperbolic paraboloid: z = x 2 2y 2 horizontal traces: hyperbolas traces parallel to xz-plane: parabolas opening up traces parallel to yz-plane: parabolas opening down saddle surface cone: z 2 = 2x 2 + 3y 2 horizontal traces: ellipses traces in vertical coordinate planes: pairs of intersecting lines ellipsoid: x 2 + 2y 2 + 3z 2 = 1 traces parallel to coordinate planes: ellipses hyperboloid of 1 sheet: x 2 + 2y 2 3z 2 = 1 horizontal traces: ellipses traces parallel to vertical coordinate planes: hyperbolas hyperboloid of 2 sheets: x 2 2y 2 + 3z 2 = 1 horizontal traces: ellipses traces parallel to vertical coordinate planes: hyperbolas opening vertically

vector function: vector limits: r continuous at a: warning: space curve: MULTIVARIABLE CALCULUS 9 14.1 Vector Functions and Space Curves r(t) = f(t), g(t), h(t) = f(t)i + g(t)j + h(t)k lim t a r(t) = lim t a f(t), lim t a g(t), lim t a h(t) lim t a r(t) = r(a) we blur the distinction between point P and position vector r = OP r = r(t) parametric equations: x = f(t), y = g(t), z = h(t)

10 JOHN QUIGG vector derivatives: tangent line to curve at P : unit tangent vector: T = r r 14.2 Derivatives and Integrals of Vector Functions dr dt = r(t + h) r(t) dx r (t) = lim = h 0 h dt, dy dt, dz dt = tangent vector to curve r = r(t) at P = (x(t), y(t), z(t)) smooth curve: r continuous and nonzero (except possibly at endpoints) line through P parallel to r (t) piecewise smooth: continuous, and smooth except at finitely many points C = C 1 C n, (C 1,..., C n smooth, overlap only at endpoints) also path properties: d dt[ u(t) + v(t) ] = u (t) + v (t) d dt[ cu(t) ] = cu (t), t R d dt[ u(f(t)) ] = f (t)u (t) (Chain Rule) Product Rules: d dt[ f(t)u(t) ] = f (t)u(t) + f(t)u (t) d dt[ u(t) v(t) ] = u (t) v(t) + u(t) v (t) d dt[ u(t) v(t) ] = u (t) v(t) + u(t) v (t) vector integral: b a r(t) dt = b a x(t) dt, b a y(t) dt, b a z(t) dt fundamental theorem: b b r (t) dt = r(t) = r(b) r(a) a a indefinite integral: r(t) dt = x(t) dt, y(t) dt, z(t) dt

MULTIVARIABLE CALCULUS 11 14.3 Arc Length arc length: L = b dr b ( a dt dt = dx ) 2 ( a dt + dy ) 2 ( dt + dz ) 2 dt dt independent of parameterization arc length function: s(t) = t dr a du du can use arc length s as parameter curvature: κ = dt T ds = r = r r r 3 (principal) unit normal vector: N = T T N perpendicular to T

12 JOHN QUIGG position at time t: velocity: v = r speed: v = v 14.4 Motion in Space: Velocity and Acceleration r(t) acceleration: a = v = r properties: v(t) = v(t 0 ) + t t 0 a(u) du r(t) = r(t 0 ) + t t 0 v(u) du v = vt Newton s Law: F = ma force=(mass)(acceleration) tangential and normal components of acceleration: a = a T T + a N N a T = v = r r r = v a v a N = κv 2 = r r r = v a v

function of 2 variables: f : D R, D R 2 z = f(x, y), (x, y) D independent variables: x, y dependent variable: z graph of f: {(x, y, z) : z = f(x, y), (x, y) D} level curves of f: {(x, y) : f(x, y) = k}, k R contour map of f: MULTIVARIABLE CALCULUS 13 15.1 Functions of Several Variables descriptive family of level curves function of 3 variables: f : D R, D R 3 w = f(x, y, z), (x, y, z) D independent variables: x, y, z dependent variable: w level surfaces: {(x, y, z) : f(x, y, z) = k}, k R

14 JOHN QUIGG 15.2 Limits and continuity limit: lim r a f(r) = L means: for all ɛ > 0 there exists δ > 0 such that f(r) L < ɛ whenever r D and 0 < r a < δ notation: f(r) L as r a test for nonexistence of limit: f(r) L 1 as r a along curve C 1 f(r) L 2 as r a along curve C 2 L 1 L 2 properties: ( ) lim f(r) + g(r) = lim f(r) + lim g(r) r a r a r a lim f(cr) = c lim f(r) r a ( ) r a lim f(r)g(r) = lim f(r) lim g(r) r a r a r a lim (x,y) (a,b) lim (x,y) (a,b) lim (x,y) (a,b) c R Squeeze Theorem: { lim f(r) = lim g(r) = L r a r a f(r) h(r) g(r) f continuous at a: continuous on D: lim r a f(r) = f(a) continuous at every a D } lim r a lim r a h(r) = L f(r) does not exist polynomial function of 2 variables: finite sum of terms of form cx n y k (c R, n, k = 0, 1, 2,... ) 3 variables: cx n y k z l (n, k, l = 0, 1, 2,... ) rational function: quotient of 2 polynomials continuous (on domain: denominator 0) fact: g(f(r)) continuous if g and f are

MULTIVARIABLE CALCULUS 15 15.3 Partial Derivatives 2 variables: z = f(x, y) traces of f: partial derivative of f with respect to x at (a, b): f f(a + h) f(a, b) (a, b) = lim = d [ ] f(x, b) x h 0 h dx notation: f x = z x = f x = x f(x, y) = f 1 { at y = b : at x = a : slopes of tangent lines to traces: f y = z y = f y = y f(x, y) = f 1 {z = f(x, b), y = b}, or r(t) = t, b, f(t, b) {z = f(a, y), x = a}, or r(t) = a, t, f(a, t) {z = f(x, b), y = b} at x = a: slope = f (a, b) x {z = f(a, y), y = b} at y = b: slope = f (a, b) y for 3 variables: w = f(x, y, z) 2nd partials: Clairaut s Theorem: higher order partials: f d [ ] (a, b, c) = f(x, b, c) x dx f x = w x f y = w y f z = w z x=a 2 f x 2 = [ ] f = f xx = ( f x x x )x 2 f y x = [ ] f = f xy = ( f x y x )y 2 f x y = [ ] f = f yx = ( f y x y )x 2 f y 2 = [ ] f = f yy = ( f y y y )y f xy, f yx continuous f xy = f yx f xyx = ( f xy ) x = 3 f x y x = f x2 zy = ( f x2 z )y = 4 f y z x 2 = x=a

16 JOHN QUIGG partial differential equations: (PDE s) Laplace s equation: 2 u x 2 + 2 u y 2 = 0 2 u 3 variables: x 2 + 2 u y 2 + 2 u z 2 = 0 Wave equation: u tt = a 2 u xx, a > 0

MULTIVARIABLE CALCULUS 17 15.4 Tangent Planes and Linear Approximations tangent plane to z = f(x, y) at (a, b, f(a, b)): plane containing tangent lines to both traces {z = f(x, y), x = a}, or r = a, y, f(a, y), at y = b {z = f(x, b), y = b}, or r = x, b, f(x, b), at x = a equation: z = f(a, y) + f x (a, b)(x a) + f y (a, b)(y b) approximates surface near (a, b, f(a, b)) linearization of f at (a, b): L(x, y) = f(a, b) + f x (a, b)(x a) + f y (a, b)(y b) linear approximation of f at (a, b): f(x, y) L(x, y), (x, y) near (a, b) also tangent plane approximation increments: x = x a, y = y b, z = f(a + x, b + y) f(a, b) f differentiable at (a, b): fact: z = f x (a, b) x + f y (a, b) y + ɛ 1 x + ɛ 2 y ɛ 1, ɛ 2 0 as ( x, y) (0, 0) f has continuous partials differentiable differentials: dx = x, dy = y, dz = f x dx + f y dy = z z dx + x y dy linear approximation f(x, y) f(a, b) + dz z dz change in z along surface approximated by change in z along tangent plane for 3 variables: w = f(x, y, z) linearization: L(x, y, z) = f(a, b, c) + f x (a, b, c)(x a) + f y (a, b, c)(y b) + f z (a, b, c)(z c) linear approximation: f(x, y, z) L(x, y, z) increment: w = f(x + x, y + y, z + z) f(x, y, z) differential: dw = w x w w dx + dy + y z dz

18 JOHN QUIGG Chain Rules: z = f(x, y), x = g(t), y = h(t) dependent variable: z intermediate variables: x, y independent variable: t z = f(x, y), x = g(s, t), y = h(s, t) dependent variable: z intermediate variables: x, y independent variables: s, t general: 15.5 The Chain Rule dz dt = z dx x dt + z dy y dt z s = z x x s + z y y s z t = z x x t + z y y t u = u(x 1,..., x n ) x j = x j (t 1,..., t m ), u n u x j =, t i x j t i j=1 dependent variable: u intermediate variables: x 1,..., x n independent variables: t 1,..., t m implicit differentiation: F (x, y) = 0 implicitly defines y = f(x) =1 {}}{ [ ] F dx F (x, y(x)) = x x F dy dx = x F y F (x, y, z) = 0 implicitly defines z = f(x, y) =1 {}}{ [ ] F x F (x, y, z(x, y)) = x x F z x = x F z F z y = y F z j = 1,..., n i = 1,..., m dx + F y x + F y dy dx = 0 =0 {}}{ y x + F z z x = 0

MULTIVARIABLE CALCULUS 19 15.6 Directional Derivatives and the Gradient Vector directional derivative of f at (x 0, y 0 ) in direction of unit vector u = a, b : f(x 0 + ha, y 0 + hb) f(x 0, y 0 ) D u f(x 0, y 0 ) = lim h 0 h = dh[ d f(x0 + ha, y 0 + hb) ] h=0 f gradient of f: grad f = f = x, f y properties: D u f(x 0, y 0 ) = f x (x 0, y 0 )a + f y (x 0, y 0 )b = grad f(x 0, y 0 ) u D u f(x 0, y 0 ) = slope of tangent line to curve at (x 0, y 0, f(x 0, y 0 )) 3 variables: D u f(x, y, z) = f u, f = properties: r(t) = x 0 + ta, y 0 + tb, f(x 0 + ta, y 0 + tb) f x, f y, f, u = a, b, c z maximum D u f at r 0 is f(r 0 ), and is in direction u = f(r 0) f(r 0 ) f(r 0 ) is perpendicular to level surface f(r) = f(r 0 ) at r 0 tangent plane to level surface is f(r 0 ) (r r 0 ) = 0 normal line to level surface f(x, y, z) = f(x 0, y 0, z 0 ) at (x 0, y 0, z 0 ) is x x 0 f x (x 0, y 0, z 0 ) = y y 0 f y (x 0, y 0, z 0 ) = z z 0 f z (x 0, y 0, z 0 ) 2 variables: f perpendicular to level curve

20 JOHN QUIGG 15.7 Maximum and Minimum Values local maximum: f(x, y) f(a, b) for all (x, y) near (a, b) (i.e., for all (x, y) in some disk centered at (a, b)) local maximum value: f(a, b) absolute maximum: f(x, y) f(a, b) for all (x, y) in domain of f similarly for minimum extreme: max or min critical point of f: both f x (a, b) = 0 or does not exist, and f y (a, b) = 0 or does not exist critical point test: f has local extreme at (a, b) (a, b) critical point notation: D = D(a, b) = f xx (a, b)f yy (a, b) f xy (a, b) 2 2nd derivative test at critical point: if 2nd-order partials continuous then: D > 0, f xx (a, b) > 0 f(a, b) local min D > 0, f xx (a, b) < 0 f(a, b) local max D < 0 f(a, b) neither local max nor local min saddle point graph of f crosses tangent plane at (a, b) D = 0: could be anything (no information) (a, b) boundary point of D: D closed: D bounded: contains all its boundary points contained in some disk Extreme Value Theorem: min on D every disk centered at (a, b) contains points in D and points outside D f continuous on closed and bounded set D f has absolute max and absolute algorithm for extremes on compact sets: when f : D R continuous and D compact: find critical points of f in D find extreme values of f on boundary of D

MULTIVARIABLE CALCULUS 21 15.8 Lagrange Multipliers constrained extremum problem: find extreme values of f(r) subject to g(r) = k fact: constrained { extreme occurs at r 0 } curves (in 2 variables) level tangent at r surfaces (in 3 variables) 0 f(r 0 ) = λ g(r 0 ) for some λ R λ = Lagrange multiplier

22 JOHN QUIGG 16.1 Double Integrals over Rectangles notation: R = [a, b] [c, d] = {(x, y) : a x b, c y d} f : R R partition [a, b]: a = x 0 < x 1 < < x m = b, x = x i x i 1, i = 1,..., m partition [c, d]: c = y 0 < y 1 < < y n = d, y = y j y j 1, j = 1,..., n ijth rectangle: R ij = [x i 1, x i ] [y j 1, y j ] area of R ij : A = x y sample point: (x ij, y ij ) R ij Riemann sum: m n i=1 j=1 f(x ij, y ij ) A double integral: R f(x, y) da = R f da = R f = lim m,n i,j f(x ij, y ij ) A fact: f 0 on R f da = volume under graph of f over R R average value: f ave = 1 A(R) R f da A(R) = area of R properties: R (f + g) = R f + R g R cf = c R f, c R f(x, y) g(x, y) for all (x, y) R R f R g

notation: f : R R, R = [a, b] [c, d] partial integral with respect to y: iterated integral: b a factored integrands: Fubini s Theorem: MULTIVARIABLE CALCULUS 23 d c f(x, y) dy dx = b a b a f(x)g(y) dx dy = b d c f continuous on R b f(x, y) da = R 16.2 Iterated Integrals A(x) = d f(x, y) dy c [ ] d c f(x, y) dy a d c a f(x) dx d c f(x, y) dy dx = dx = b A(x) dx a g(y) dy d b c a f(x, y) dx dy

24 JOHN QUIGG notation: f : D R, D R 2 find rectangle R containing D define F : R R by double integral of f over D: type I region: type II region: 16.3 Double Integrals over General Regions F (x, y) = { f(x, y) (x, y) D 0 (x, y) / D D f(x, y) da = D f da = D f = R F D = {(x, y) : a x b, g 1 (x) y g 2 (x)} b g2(x) f da = f(x, y) dy dx D a g 1(x) D = {(x, y) : c y d, h 1 (y) x h 2 (y)} d h2(y) f da = f(x, y) dx dy D properties: D (f + g) = D f + D g D cf = c D f, c R f(x, y) g(x, y) for all (x, y) D D f D g D = D 1 D 2, D 1, D 2 overlap only on their boundaries D f = D 1 f + D 2 f D 1 = A(D) m f(x, y) M for all (x, y) D ma(d) D f MA(D) c h 1(y)

MULTIVARIABLE CALCULUS 25 16.4 Double Integrals in Polar Coordinates polar rectangle: R = {(r, θ) : a r b, α θ β} A(R) = r r θ r = r ave = a+b 2, r = b a, θ = β α double integrals in polar coordinates: da = r dr dθ b a R f(x, y) da = β f(r cos θ, r sin θ)r dr dθ α D = {(r, θ) : α θ β, h 1 (θ r h 2 (θ)} D f(x, y) da = β h2(θ) α h 1(θ) f(r cos θ, r sin θ)r dr dθ

26 JOHN QUIGG thin plate ( lamina ) occupying region D R 2 density: negligible thickness mass/(unit area) 16.5 Applications of Double Integrals m ρ(x, y) = lim x, y 0 A A = x y = area of rectangle containing (x, y) m = mass of plate inside rectangle mass: m = D ρ da center of mass: (x, y) x = 1 xρ(x, y) da m D y = 1 yρ(x, y) da m D centroid if density ρ constant plate balances on knife edge at x, also on knife edge at y, and on knife point at (x, y) = weighted average of x = weighted average of y

notation: MULTIVARIABLE CALCULUS 27 16.7 Triple Integrals B = [a, b] [c, d] [r, s] = {(x, y, z) : a x b, c y d, r z s} f : B R a = x 0 < x 1 < < x n = b, x = x i x i 1, i = 1,..., l c = y 0 < y 1 < < y m = d, y = y j y j 1, j = 1,..., m r = z 0 < z 1 < < z p = s, z = z k z k 1, k = 1,..., n ijkth box: B ijk = [x i 1, x i ] [y j 1, y j ] [z k 1, z k ] volume of B ijk : V = x y z sample point: (x ijk, y ijk, z ijk ) B ijk Riemann sum: l m n i=1 j=1 k=1 f(x ijk, y ijk, z ijk ) V triple integral: Fubini s Theorem: factored integrands: B f(x, y, z) dv = B f dv = B f = lim l,m,n f continuous on B f(x, y, z) dv = s r notation: f : E R, E R 3 find box B containing E define F : B R by triple integral of f over E: type 1 region: B = s d b r c a s b d r a c f(x, y, z) dx dy dz f(x, y, z) dy dx dz = (6 possibilities) d b c a f(x)g(y)h(z) dx dy dz = b a f(x) dx d F (x, y, z) = { f(x, y, z) (x, y, z) E 0 (x, y, z) / E i,j,k f(x ij, y ij, z ijk ) V c g(y) dy s h(z) dz E f(x, y, z) dv = E f dv = E f = B F E = {(x, y, z) : (x, y) D, u 1 (x, y) z u 2 (x, y)}, D R 2 [ ] u2(x,y) f dv = f(x, y, z) dz da E D u 1(x,y) r type 2 region: type 3 region: E = {(x, y, z) : (y, z) D, u 1 (y, z) x u 2 (y, z)}, D R 2 [ ] u2(y,z) f dv = f(x, y, z) dx da E D u 1(y,z) E = {(x, y, z) : (x, z) D, u 1 (x, z) y u 2 (x, z)}, D R 2 [ ] u2(x,z) f dv = f(x, y, z) dy da E D u 1(x,z) properties: Similar to double integrals. In particular, 1 dv = V (E) = volume of E E

28 JOHN QUIGG density: mass/(unit volume) m ρ(x, y, z) = lim x, y, z 0 V V = x y z = volume of box containing (x, y, z) m = mass of solid inside box mass: m = E ρ dv center of mass: (x, y, z) x = 1 m y = 1 m z = 1 m centroid if density ρ constant E E E xρ(x, y, z) dv yρ(x, y, z) dv zρ(x, y, z) dv

MULTIVARIABLE CALCULUS 29 13.7 Cylindrical and Spherical Coordinates cylindrical coordinates: (r, θ, z) (r, θ) = polar coordinates of projection of point into xy-plane z = usual 3rd Cartesian (rectangular) coordinate properties: x = r cos θ, y = r sin θ, z = z r 2 = x 2 + y 2 tan θ = y x r = r 0 is cylinder centered on z-axis with radius r 0 θ = θ 0 is plane containing z-axis making angle θ 0 with half-plane {y = 0, x 0} spherical coordinates: (ρ, θ, φ) ρ = distance from point to origin θ = usual 2nd cylindrical coordinate φ = angle from positive z-axis to line segment from origin to point properties: r = ρ sin φ (r = usual 1st cylindrical coordinate) x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ ρ 2 = x 2 + y 2 + z 2 ρ = ρ 0 is sphere centered at origin with radius ρ 0 θ = θ 0 is half-plane with edge at z-axis making angle θ 0 with half-plane {y = 0, x 0} φ = φ 0 is half-cone with vertex at origin, centered on positive z-axis, making angle φ 0 with positive z-axis

30 JOHN QUIGG 16.8 Integrals in Cylindrical and Spherical Coordinates triple integrals in cylindrical coordinates: f : E R, E type 1 [ ] u2(x,y) f(x, y, z) dv = f(x, y, z) dz da E = D u 1(x,y) β h2(θ) u2(r cos θ,r sin θ) α h 1(θ) u 1(r cos θ,r sin θ) f(r cos θ, r sin θ, z)r dz dr dθ 1st integrate with respect to z, then do outer double integral in polar coordinates spherical wedge: E = {(ρ, θ, φ) : a ρ b, α θ β, c φ d} V (E) = ρ 2 sin φ ρ θ φ, for some (ρ, θ, φ) E triple integrals in spherical coordinates: over spherical wedge: d β b f(x, y, z) dv = f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ 2 sin φ dρ dθ dφ E more general region: for example, c α a E = {(ρ, θ, φ) : α θ β, c φ d, g 1 (θ, φ) ρ g 2 (θ, φ)} d β g2(θ,φ) f(x, y, z) dv = f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ 2 sin φ dρ dθ dφ E c α g 1(θ,φ)

2-d vector field: F: D (2-d vectors), D R 2 MULTIVARIABLE CALCULUS 31 17.1 Vector Fields F(x, y) = P (x, y)i + Q(x, y)j P, Q: D R component functions scalar field: real-valued function of several variables 3-d vector field: F: E (3-d vectors), E R 3 F continuous: applications: gradient field: F = P i + Qj + Rk P, Q, R: E R component functions component functions continuous velocity field, force field f = f x i + f y j + f z k F conservative: F = f for some f f potential function for F

32 JOHN QUIGG line integral of f along smooth curve C: f(x, y) ds = C : r = r(t), a t b independent of parameterization of C notation: ds = r (t) dt similar in 2 or 3 dimensions C 17.2 Line Integrals C f ds = b a f(r(t)) r (t) dt along path (piecewise smooth curve): C = C 1 C n, (C 1,..., C n smooth): n f ds = f ds C i=1 C i arc length: C 1 ds application: C wire density: ρ = mass/(unit of arc length) mass: m = C ρ ds center of mass: (x, y, z) x = 1 xρ(x, y, z) ds m C y = 1 yρ(x, y, z) ds m C z = 1 zρ(x, y, z) ds m oriented curve: choose continuous field of unit tangent vectors parameterization gives orientation: T = r / r line integral of (continuous) vector field along oriented curve: b F dr = F T ds = F(r(t)) r (t) dt C notation: F T = tangential component of F dr = T ds = dr dt dt F = P i + Qj + Rk C F dr = P dx + Q dy + R dz C C P dx = b P (x(t), y(t), z(t))dx a dt dt,... notation: C = C with opposite orientation properties: C F dr = C F dr (depends upon orientation) C f ds = C f ds (independent of orientation) C C a application: work done by force field F in moving particle along curve C: W = C F dr

MULTIVARIABLE CALCULUS 33 17.3 The Fundamental Theorem for Line Integrals Fundamental Theorem of Line Integrals: C : r = r(t) (a t b) smooth, f continuous b f dr = f(r(t)) = f(r(b)) f(r(a)) C line integrals of F independent of path in D: for all C 1, C 2 paths in D with same initial and final points, F dr = C 1 F dr C 2 closed path: initial and final points coincide r(a) = r(b) fact: C F dr independent of path in D F dr = 0 for every closed path in D C D open: comprises interior points only D connected: fact: fact: any 2 points in D can be joined by path in D ( D is all in one piece ) F continuous on open connected set D the following are equivalent: line integrals of F independent of path in D F conservative in D simple curve: F(x, y) = P (x, y)i + Q(x, y)j conservative P, Q have continuous partials D simply connected: no holes fact: a P y = Q x r(a) = r(b), but r(t 1 ) r(t 2 ) whenever a t 1 < t 2 < b connected, and every simple curve in D is the boundary of a set contained in D F(x, y) = P (x, y)i + Q(x, y)j on open simply connected set D P, Q have continuous partials P y = Q x F conservative

34 JOHN QUIGG notation: F = P i + Qj + Rk 17.5 Curl and Divergence curl of F: curl F = F = (R y Q z )i + (P z R x )j + (Q x P y )k think: = x i + y j + z k properties: f has continuous 2nd partials curl grad f = 0 F conservative and has continuous partials curl F = 0 conversely if domain F = R 3 F = velocity field curl F measures tendency of fluid to rotate divergence of F: div F = F = P x + Q y + R z properties: F has continuous 2nd partials div curl F = 0 F = velocity field div F measures tendency of fluid to expand Laplacian: 2 f = f = f xx + f yy + f zz

MULTIVARIABLE CALCULUS 35 17.6 Parametric Surfaces parameterized surface: r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k (u, v) D R 2 u, v parameters {x = x(u, v), y = y(u, v), dz = z(u, v)} parametric equations of surface { r(u, v 0 ) (v constant) grid curves: r(u 0, v) (u constant) r u = x u i + y u j + z u k tangent to r(u, v 0) tangent vectors to grid curves: r v = x v i + y v j + z v k tangent to r(u 0, v) all partials evaluated at (u 0, v 0 ) smooth surface: r u r v continuous and nonzero tangent plane to smooth surface: r = r(u 0, v 0 ) + ur u (u 0, v 0 ) + vr v (u 0, v 0 ) normal vector r u r v surface area: A(S) = D r u r v da S : r = r(u, v), (u, v) D R 2 special cases: graph of z = f(x, y): r = xi + yj + f(x, y)k, r x = i + f x k r y = j + f y k r x r y = f x i f y j + k A(graph) = fx 2 + fy 2 + 1 da D (x, y) D surface of revolution: rotate y = f(x) (a x b) about x-axis r = xi + f(x) cos θj + f(x) sin θk r x = i + f (x) cos θj + f (x) sin θk r θ = f(x) sin θj + f(x) cos θk r x r θ = f(x) ( f (x)i cos θj sin θk ) A(S) = b 2π = 2π a 0 b a a x b, 0 θ 2π f(x) f (x)i cos θj sin θk dθ dx f(x) f (x) 2 + 1 dx

36 JOHN QUIGG 17.7 Surface Integrals surface integral of f over smooth surface S: f(x, y, z) ds = f ds = S S : r = r(u, v), (u, v) D independent of parameterization of S notation ds = r u r v da over piecewise smooth surface: fact: A(S) = S 1 ds oriented surface: choose continuous field of unit normal vectors parameterization gives orientation: n = S D f(r(u, v)) r u r v da S = S 1 S n, (S 1,..., S n smooth, overlap only on boundaries): n f ds = f ds S S i i=1 ru rv r u r v closed surface: boundary of solid region E positive orientation: outward unit normals surface integral of (continuous) vector field over oriented surface: F ds = F n ds = F(r(u, v)) (r u r v ) da S S = flux of F across S notation: F n = normal component of F ds = n ds = r u r v da F = velocity field F ds = volume of fluid flowing across S per unit time S D

notation: MULTIVARIABLE CALCULUS 37 17.8 Stokes Theorem S oriented piecewise smooth surface with boundary simple closed path C positive orientation of C: as unit normal n traverses C, S is on left Stokes Theorem: S oriented piecewise smooth surface with positively oriented boundary curve C F continuous partials on open set in R 3 containing S F ds = F dr special case: S = D xy-plane, F = P (x, y)i + Q(x, y)j S r = xi + yj, r x = i, C n = k r y = j r x r y = k, F = (Q x P y )k Green s Theorem: (Q x P y ) da = P dx + Qdy D C

38 JOHN QUIGG 17.9 Divergence Theorem Divergence Theorem: E solid region in R 3 with positively oriented piecewise smooth boundary surface S F continuous partials on open set in R 3 containing E F dv = F ds E S