Calculus 3 formula sheet, Cheat Sheet of Calculus

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Harvard College Math 21a: Multivariable Calculus Formula and Theorem Review 1 Contents Table of Contents 4 9 Vectors and the Geometry of Space 5 9.1 Distance Formula in 3 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 5 9.2 Equation of a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9.3 Properties of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9.4 Unit Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9.5 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9.6 Properties of the Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . 5 9.7 Vector Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.8 Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.9 Properties of the Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.10 Scalar Triple Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.11 Vector Equation of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.12 Symmetric Equations of a Line . . . . . . . . . . . . . . . . . . . . . . . . . 6 9.13 Segment of a Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9.14 Vector Equation of a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9.15 Scalar Equation of a Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 9.16 Distance Between Point and Plane . . . . . . . . . . . . . . . . . . . . . . . 7 9.17 Distance Between Point and Line . . . . . . . . . . . . . . . . . . . . . . . . 7 9.18 Distance Between Line and Line . . . . . . . . . . . . . . . . . . . . . . . . . 7 9.19 Distance Between Plane and Plane . . . . . . . . . . . . . . . . . . . . . . . 8 9.20 Quadric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9.21 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9.22 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10 Vector Functions 9 10.1 Limit of a Vector Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.2 Derivative of a Vector Function . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.3 Unit Tangent Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.4 Derivative Rules for Vector Functions . . . . . . . . . . . . . . . . . . . . . . 9 10.5 Integral of a Vector Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.6 Arc Length of a Vector Function . . . . . . . . . . . . . . . . . . . . . . . . . 9 10.7 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.8 Normal and Binormal Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.9 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10.10Parametric Equations of Trajectory . . . . . . . . . . . . . . . . . . . . . . . 10 10.11Tangential and Normal Components of Acceleration . . . . . . . . . . . . . . 10 10.12Equations of a Parametric Surface . . . . . . . . . . . . . . . . . . . . . . . . 10 2 9 Vectors and the Geometry of Space 9.1 Distance Formula in 3 Dimensions The distance between the points P1(x1, y1, z1) and P2(x2, y2, z2) is given by: |P1P2| = √ (x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2 9.2 Equation of a Sphere The equation of a sphere with center (h, k, l) and radius r is given by: (x− h)2 + (y − k)2 + (z − l)2 = r2 9.3 Properties of Vectors If ~a,~b, and ~c are vectors and c and d are scalars: ~a+~b = ~b+ ~a ~a+ 0 = ~a ~a+ (~b+ ~c) = (~a+~b) + ~c ~a+−~a = 0 c(~a+~b) = c~a+ c+~b (c+ d)~a = c~a+ d~a (cd)~a = c(d~a) 9.4 Unit Vector A unit vector is a vector whose length is 1. The unit vector ~u in the same direction as ~a is given by: ~u = ~a |~a| 9.5 Dot Product ~a ·~b = |~a||~b| cos θ ~a ·~b = a1b1 + a2b2 + a3b3 9.6 Properties of the Dot Product Two vectors are orthogonal if their dot product is 0. ~a · ~a = |~a|2 ~a ·~b = ~b · ~a ~a · (~b+ ~c) = ~a ·~b+ ~a · ~c (c~a) ·~b = c(~a ·~b) = ~a · (c~b) 0 · ~a = 0 5 9.7 Vector Projections Scalar projection of ~b onto ~a: comp~a ~b = ~a ·~b |~a| Vector projection of ~b onto ~a: proj~a ~b = ( ~a ·~b |~a| ) ~a |~a| 9.8 Cross Product ~a×~b = (|~a||~b| sin θ)~n where ~n is the unit vector orthogonal to both ~a and ~b. ~a×~b = 〈a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1〉 9.9 Properties of the Cross Product Two vectors are parallel if their cross product is 0. ~a×~b = −~b× ~a (c~a)×~b = c(~a×~b) = ~a× (c~b) ~a× (~b+ ~c) = ~a×~b+ ~a× ~c (~a+~b)× ~c = ~a× ~c+~b× ~c 9.10 Scalar Triple Product The volume of the parallelpiped determined by vectors ~a, ~b, and ~c is the magnitude of their scalar triple product: V = |~a · (~b× ~c)| ~a · (~b× ~c) = ~c · (~a×~b) 9.11 Vector Equation of a Line ~r = ~r0 + t~v 9.12 Symmetric Equations of a Line x− x0 a = y − y0 b = z − z0 c where the vector ~c = 〈a, b, c〉 is the direction of the line. The symmetric equations for a line passing through the points (x0, y0, z0) and (x1, y1, z1) are given by: x− x0 x1 − x0 = y − y0 y1 − y0 = z − z0 z1 − z0 6 9.13 Segment of a Line The line segment from ~r0 to ~r1 is given by: ~r(t) = (1− t)~r0 + t~r1 for 0 ≤ t ≤ 1 9.14 Vector Equation of a Plane ~n · (~r − ~r0) = 0 where ~n is the vector orthogonal to every vector in the given plane and ~r − ~r0 is the vector between any two points on the plane. 9.15 Scalar Equation of a Plane a(x− x0) + b(y − y0) + c(z − z0) = 0 where (x0, y0, z0) is a point on the plane and 〈a, b, c〉 is the vector normal to the plane. 9.16 Distance Between Point and Plane D = |ax1 + by1 + cz1 + d|√ a2 + b2 + c2 d(P,Σ) = | ~PQ · ~n| |~n| where P is a point, Σ is a plane, Q is a point on plane Σ, and ~n is the vector orthogonal to the plane. 9.17 Distance Between Point and Line d(P,L) = | ~PQ× ~u| |~u| where P is a point in space, Q is a point on the line L, and ~u is the direction of line. 9.18 Distance Between Line and Line d(L,M) = |( ~PQ) · (~u× ~v)| |~u× ~v| where P is a point on line L, Q is a point on line M , ~u is the direction of line L, and ~v is the direction of line M . 7 10.7 Curvature κ = |d ~T ds | = | ~T ′(t)| |~r ′(t)| κ = |~r ′(t)× ~r ′′(t)| |~r ′(t)|3 κ(x) = |f ′′(x)| [1 + (f ′(x))2]3/2 10.8 Normal and Binormal Vectors ~N(t) = ~T ′(t) |~T ′(t)| ~B(t) = ~T (t)× ~N(t) 10.9 Velocity and Acceleration ~v(t) = ~r ′(t) ~a(t) = ~v ′(t) = ~r ′′(t) 10.10 Parametric Equations of Trajectory x = (v0 cosα)t y = (v0 sinα)t− 1 2 gt2 10.11 Tangential and Normal Components of Acceleration ~a = v′ ~T + κv2 ~N 10.12 Equations of a Parametric Surface x = x(u, v) y = y(u, v) z = z(u, v) 11 Partial Derivatives 11.1 Limit of f(x, y) If f(x, y)→ L1 as (x, y)→ (a, b) along a path C1 and f(x, y)→ L2 as (x, y)→ (a, b) along a path C2, then lim(x,y)→(a,b) f(x, y) does not exist. 10 11.2 Strategy to Determine if Limit Exists 1. Substitute in for x and y. If point is defined, limit exists. If not, continue. 2. Approach (x, y) from the x-axis by setting y = 0 and taking limx→a. Compare this result to approaching (x, y) from the y-axis by setting x = 0 and taking limy→a. If these results are different, then the limit does not exist. If results are the same, continue. 3. Approach (x, y) from any nonvertical line by setting y = mx and taking limx→a. If this limit depends on the value of m, then the limit of the function does not exist. If not, continue. 4. Rewrite the function in cylindrical coordinates and take limr→a. If this limit does not exist, then the limit of the function does not exist. 11.3 Continuity A function is continuous at (a, b) if lim (x,y)→(a,b) f(x, y) = f(a, b) 11.4 Definition of Partial Derivative fx(a, b) = g′(a) where g(x) = f(x, b) fx(a, b) = lim h→0 f(a+ h, b)− f(a, b) h To find fx, regard y as a constant and differentiate f(x, y) with respect to x. 11.5 Notation of Partial Derivative fx(x, y) = fx = ∂f ∂x = ∂ ∂x f(x, y) = Dxf 11.6 Clairaut’s Theorem If the functions fxy and fyx are both continuous, then fxy(a, b) = fyx(a, b) 11.7 Tangent Plane z − z0 = fx(x0, y0)(x− x0) + fy(x0, y0)(y − y0) 11 11.8 The Chain Rule dz dt = ∂z ∂x dx dt + ∂z ∂y dy dt 11.9 Implicit Differentiation dy dx = − ∂F ∂x ∂F ∂y 11.10 Gradient ∇f(x, y) = 〈fx(x, y), fy(x, y)〉 11.11 Directional Derivative D~uf(x, y) = ∇f(x, y) · ~u where ~u = 〈a, b〉 is a unit vector. 11.12 Maximizing the Directional Derivative The maximum value of the directional derivative D~uf(x) is |∇f(x)| and it occurs when ~u has the same direction as the gradient vector ∇f(x). 11.13 Second Derivative Test Let D = fxx(a, b)fyy(a, b)− (fxy(a, b)) 2. 1. If D > 0 and fxx(a, b) > 0 then f(a, b) is a local minimum. 2. If D > 0 and fxx(a, b) < 0 then f(a, b) is a local maximum. 3. If D < 0 and fxx(a, b) > 0 then f(a, b) is a not a local maximum or minimum, but could be a saddle point. 11.14 Method of Lagrange Multipliers To find the maximum and minimum values of f(x, y, z) subject to the constraint g(x, y, z) = k: 1. Find all values of x, y, z and λ such that ∇f(x, y, z) = λ∇g(x, y, z) and g(x, y, z) = k 2. Evaluate f at all of these points. The largest is the maximum value, and the smallest is the minimum value of f subject to the constraint g. 12 13.9 Flux ∫∫ S ~F · d~S = ∫∫ D ~F · (~ru × ~rv) dA 13.10 Stokes’ Theorem∫ C ~F · d~r = ∫∫ S curl(~F ) · d~S 13.11 Divergence Theorem∫∫ S ~F · d~S = ∫∫∫ E div(~F ) dV 14 Appendix A: Selected Surface Paramatrizations 14.1 Sphere of Radius ρ ~r(u, v) = 〈ρ cosu sin v, ρ sinu sin v, ρ cos v〉 14.2 Graph of a Function f(x, y) ~r(u, v) = 〈u, v, f(u, v)〉 14.3 Graph of a Function f(φ, r) ~r(u, v) = 〈v cosu, v sinu, f(u, v)〉 14.4 Plane Containing P, ~u, and ~v ~r(s, t) = ~OP + s~u+ t~v 14.5 Surface of Revolution ~r(u, v) = 〈g(v) cosu, g(v) sinu, v〉 where g(z) gives the distance from the z-axis. 14.6 Cylinder ~r(u, v) = 〈cosu, sinu, v〉 15 14.7 Cone ~r(u, v) = 〈v cosu, v sinu, v〉 14.8 Paraboloid ~r(u, v) = 〈 √ v cosu, √ v sinu, v〉 15 Appendix B: Selected Differential Equations 15.1 Heat Equation ft = fxx 15.2 Wave Equation (Wavequation) ftt = fxx 15.3 Transport (Advection) Equation fx = ft 15.4 Laplace Equation fxx = −fyy 15.5 Burgers Equation fxx = ft + ffx 16