Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review

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1 Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 December 15,

2 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce Distnce Formul in 3 Dimensions Eqution of Sphere Properties of Vectors Unit Vector Dot Product Properties of the Dot Product Vector Projections Cross Product Properties of the Cross Product Sclr Triple Product Vector Eqution of Line Symmetric Equtions of Line Segment of Line Vector Eqution of Plne Sclr Eqution of Plne Distnce Between Point nd Plne Distnce Between Point nd Line Distnce Between Line nd Line Distnce Between Plne nd Plne Qudric Surfces Cylindricl Coordintes Sphericl Coordintes Vector Functions Limit of Vector Function Derivtive of Vector Function Unit Tngent Vector Derivtive Rules for Vector Functions Integrl of Vector Function Arc Length of Vector Function Curvture Norml nd Binorml Vectors Velocity nd Accelertion Prmetric Equtions of Trjectory Tngentil nd Norml Components of Accelertion Equtions of Prmetric Surfce

3 11 Prtil Derivtives Limit of f(x, y) Strtegy to Determine if Limit Exists Continuity Definition of Prtil Derivtive Nottion of Prtil Derivtive Clirut s Theorem Tngent Plne The Chin Rule Implicit Differentition Grdient Directionl Derivtive Mximizing the Directionl Derivtive Second Derivtive Test Method of Lgrnge Multipliers Multiple Integrls Volume under Surfce Averge Vlue of Function of Two Vribles Fubini s Theorem Splitting Double Integrl Double Integrl in Polr Coordintes Surfce Are Surfce Are of Grph Triple Integrls in Sphericl Coordintes Vector Clculus Line Integrl Fundmentl Theorem of Line Integrls Pth Independence Curl Conservtive Vector Field Test Divergence Green s Theorem Surfce Integrl Flux Stokes Theorem Divergence Theorem Appendix A: Selected Surfce Prmtriztions Sphere of Rdius ρ Grph of Function f(x, y) Grph of Function f(φ, r)

4 14.4 Plne Contining P, u, nd v Surfce of Revolution Cylinder Cone Prboloid Appendix B: Selected Differentil Equtions Het Eqution Wve Eqution (Wveqution) Trnsport (Advection) Eqution Lplce Eqution Burgers Eqution

5 9 Vectors nd the Geometry of Spce 9.1 Distnce Formul in 3 Dimensions The distnce between the points P 1 (x 1, y 1, z 1 ) nd P 2 (x 2, y 2, z 2 ) is given by: P 1 P 2 = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) Eqution of Sphere The eqution of sphere with center (h, k, l) nd rdius r is given by: (x h) 2 + (y k) 2 + (z l) 2 = r Properties of Vectors If, b, nd c re vectors nd c nd d re sclrs: 9.4 Unit Vector + b = b = + ( b + c) = ( + b) + c + = 0 c( + b) = c + c + b (c + d) = c + d (cd) = c(d ) A unit vector is vector whose length is 1. The unit vector u in the sme direction s is given by: u = 9.5 Dot Product b = b cos θ b = 1 b b b Properties of the Dot Product Two vectors re orthogonl if their dot product is 0. = 2 b = b ( b + c) = b + c (c ) b = c( b) = (c b) 0 = 0 5

6 9.7 Vector Projections Sclr projection of b onto : Vector projection of b onto : comp b = b ( ) proj b b = 9.8 Cross Product b = ( b sin θ) n where n is the unit vector orthogonl to both nd b. b = 2 b 3 3 b 2, 3 b 1 1 b 3, 1 b 2 2 b Properties of the Cross Product Two vectors re prllel if their cross product is 0. b = b (c ) b = c( b) = (c b) ( b + c) = b + c ( + b) c = c + b c 9.10 Sclr Triple Product The volume of the prllelpiped determined by vectors, b, nd c is the mgnitude of their sclr triple product: V = ( b c) 9.11 Vector Eqution of Line ( b c) = c ( b) r = r 0 + t v 9.12 Symmetric Equtions of Line x x 0 = y y 0 = z z 0 b c where the vector c =, b, c is the direction of the line. The symmetric equtions for line pssing through the points (x 0, y 0, z 0 ) nd (x 1, y 1, z 1 ) re given by: x x 0 = y y 0 = z z 0 x 1 x 0 y 1 y 0 z 1 z 0 6

7 9.13 Segment of Line The line segment from r 0 to r 1 is given by: r(t) = (1 t) r 0 + t r 1 for 0 t Vector Eqution of Plne n ( r r 0 ) = 0 where n is the vector orthogonl to every vector in the given plne nd r r 0 is the vector between ny two points on the plne Sclr Eqution of Plne (x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 where (x 0, y 0, z 0 ) is point on the plne nd, b, c is the vector norml to the plne Distnce Between Point nd Plne D = x 1 + by 1 + cz 1 + d 2 + b 2 + c 2 d(p, Σ) = P Q n n where P is point, Σ is plne, Q is point on plne Σ, nd n is the vector orthogonl to the plne Distnce Between Point nd Line d(p, L) = P Q u u where P is point in spce, Q is point on the line L, nd u is the direction of line Distnce Between Line nd Line d(l, M) = ( P Q) ( u v) u v where P is point on line L, Q is point on line M, u is the direction of line L, nd v is the direction of line M. 7

8 9.19 Distnce Between Plne nd Plne d = e d n where n is the vector orthogonl to both plnes, e is the constnt of one plne, nd d is the constnt of the other. The distnce between non-prllel plnes is Qudric Surfces Ellipsoid: Elliptic Prboloid: Hyperbolic Prboloid: Cone: Hyperboloid of One Sheet: Hyperboloid of Two Sheets: x 2 + y2 2 b + z2 2 c = 1 2 z c = x2 + y2 2 b 2 z c = x2 y2 2 b 2 z 2 c = x2 2 + y2 2 b 2 x 2 + y2 2 b z2 2 c = 1 2 x2 2 y2 b 2 + z2 c 2 = Cylindricl Coordintes To convert from cylindricl to rectngulr: x = r cos θ y = r sin θ z = z To convert from rectngulr to cylindricl: r 2 = x 2 + y 2 tn θ = y x z = z 9.22 Sphericl Coordintes To convert from sphericl to rectngulr: x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ To convert from rectngulr to sphericl: ρ 2 = x 2 + y 2 + z 2 tn θ = y x cos φ = z ρ 8

9 10 Vector Functions 10.1 Limit of Vector Function lim r(t) = t lim t f(t), lim t g(t), lim t h(t) 10.2 Derivtive of Vector Function d r dt = r r(t + h) r(t) (t) = lim h 0 h r (t) = f (t), g (t), h (t) 10.3 Unit Tngent Vector T (t) = r (t) r (t) 10.4 Derivtive Rules for Vector Functions d dt [ u(t) + v(t)] = u (t) + v (t) d dt [c u(t)] = c u (t) 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) d dt [ u(f(t))] = f (t) u (f(t)) 10.5 Integrl of Vector Function b b r(t) dt = f(t) dt, b 10.6 Arc Length of Vector Function L = b r (t) dt g(t) dt, b h(t) dt 9

10 10.7 Curvture κ = d T ds = T (t) r (t) κ = r (t) r (t) r (t) 3 κ(x) = f (x) [1 + (f (x)) 2 ] 3/ Norml nd Binorml Vectors N(t) = T (t) T (t) B(t) = T (t) N(t) 10.9 Velocity nd Accelertion v(t) = r (t) (t) = v (t) = r (t) Prmetric Equtions of Trjectory x = (v 0 cos α)t y = (v 0 sin α)t 1 2 gt Tngentil nd Norml Components of Accelertion = v T + κv 2 N Equtions of Prmetric Surfce x = x(u, v) y = y(u, v) z = z(u, v) 11 Prtil Derivtives 11.1 Limit of f(x, y) If f(x, y) L 1 s (x, y) (, b) long pth C 1 nd f(x, y) L 2 s (x, y) (, b) long pth C 2, then lim (x,y) (,b) f(x, y) does not exist. 10

11 11.2 Strtegy to Determine if Limit Exists 1. Substitute in for x nd y. If point is defined, limit exists. If not, continue. 2. Approch (x, y) from the x-xis by setting y = 0 nd tking lim x. Compre this result to pproching (x, y) from the y-xis by setting x = 0 nd tking lim y. If these results re different, then the limit does not exist. If results re the sme, continue. 3. Approch (x, y) from ny nonverticl line by setting y = mx nd tking lim x. If this limit depends on the vlue of m, then the limit of the function does not exist. If not, continue. 4. Rewrite the function in cylindricl coordintes nd tke lim r. If this limit does not exist, then the limit of the function does not exist Continuity A function is continuous t (, b) if lim f(x, y) = f(, b) (x,y) (,b) 11.4 Definition of Prtil Derivtive f x (, b) = g () where g(x) = f(x, b) f x (, b) = lim h 0 f( + h, b) f(, b) h To find f x, regrd y s constnt nd differentite f(x, y) with respect to x Nottion of Prtil Derivtive 11.6 Clirut s Theorem f x (x, y) = f x = f x = x f(x, y) = D xf If the functions f xy nd f yx re both continuous, then 11.7 Tngent Plne f xy (, b) = f yx (, b) z z 0 = f x (x 0, y 0 )(x x 0 ) + f y (x 0, y 0 )(y y 0 ) 11

12 11.8 The Chin Rule dz dt = z dx x dt + z dy y dt 11.9 Implicit Differentition F dy dx = x F y Grdient f(x, y) = f x (x, y), f y (x, y) Directionl Derivtive where u =, b is unit vector. D u f(x, y) = f(x, y) u Mximizing the Directionl Derivtive The mximum vlue of the directionl derivtive D u f(x) is f(x) nd it occurs when u hs the sme direction s the grdient vector f(x) Second Derivtive Test Let D = f xx (, b)f yy (, b) (f xy (, b)) If D > 0 nd f xx (, b) > 0 then f(, b) is locl minimum. 2. If D > 0 nd f xx (, b) < 0 then f(, b) is locl mximum. 3. If D < 0 nd f xx (, b) > 0 then f(, b) is not locl mximum or minimum, but could be sddle point Method of Lgrnge Multipliers To find the mximum nd minimum vlues of f(x, y, z) subject to the constrint g(x, y, z) = k: 1. Find ll vlues of x, y, z nd λ such tht f(x, y, z) = λ g(x, y, z) nd g(x, y, z) = k 2. Evlute f t ll of these points. The lrgest is the mximum vlue, nd the smllest is the minimum vlue of f subject to the constrint g. 12

13 12 Multiple Integrls 12.1 Volume under Surfce V = f(x, y) dx dy 12.2 Averge Vlue of Function of Two Vribles 12.3 Fubini s Theorem R f(x, y) da = D f vg = 1 A(R) b d c R f(x, y) dx dy f(x, y) dy dx = d b c f(x, y) dx dy 12.4 Splitting Double Integrl g(x)h(y) da = b R g(x) dx d c h(y) dy 12.5 Double Integrl in Polr Coordintes f(x, y) da = b d c R f(r cos θ, r sin θ)r dr dθ 12.6 Surfce Are A(S) = r u r v da where smooth prmetric surfce S is given by r(u, v) = x(u, v), y(u, v), z(u, v). D 12.7 Surfce Are of Grph A(S) = D 1 + ( ) 2 z + x ( ) 2 z y 13

14 12.8 Triple Integrls in Sphericl Coordintes f(x, y, z) dv = d β b c α E 13 Vector Clculus f(ρ sin φ cos θ, ρ sin φ sin θ, ρ cos φ)ρ 2 sin φ dρ dθ dφ 13.1 Line Integrl C F d r = b F ( r(t)) r (t)dt 13.2 Fundmentl Theorem of Line Integrls C f d r = f( r(b)) f( r()) 13.3 Pth Independence C F d r is independent of pth in D if nd only if C F d r = 0 for every closed pth C in D Curl curl( F ) = F 13.5 Conservtive Vector Field Test F is conservtive if curl F = 0 nd the domin is closed nd simply connected Divergence div( F ) = F 13.7 Green s Theorem C F d r = R curl( F ) dx dy 13.8 Surfce Integrl f(x, y, z) ds = f( r(u, v)) r u r v da S D 14

15 13.9 Flux F ds = F ( r u r v ) da S D Stokes Theorem C F d r = S curl( F ) d S Divergence Theorem F ds = div( F ) dv S E 14 Appendix A: Selected Surfce Prmtriztions 14.1 Sphere of Rdius ρ r(u, v) = ρ cos u sin v, ρ sin u sin v, ρ cos v 14.2 Grph of Function f(x, y) r(u, v) = u, v, f(u, v) 14.3 Grph of Function f(φ, r) r(u, v) = v cos u, v sin u, f(u, v) 14.4 Plne Contining P, u, nd v 14.5 Surfce of Revolution where g(z) gives the distnce from the z-xis. r(s, t) = OP + s u + t v r(u, v) = g(v) cos u, g(v) sin u, v 14.6 Cylinder r(u, v) = cos u, sin u, v 15

16 14.7 Cone 14.8 Prboloid r(u, v) = v cos u, v sin u, v r(u, v) = v cos u, v sin u, v 15 Appendix B: Selected Differentil Equtions 15.1 Het Eqution f t = f xx 15.2 Wve Eqution (Wveqution) f tt = f xx 15.3 Trnsport (Advection) Eqution 15.4 Lplce Eqution 15.5 Burgers Eqution f x = f t f xx = f yy f xx = f t + ff x 16