Cheat Sheet for FINAL-M252-Calculus III - Fall 2022 - Key multivariable calculus formulas

Q: What should you do if a constant (e.g., α) is not defined in the problem?

Do not assign a value. Treat it as a fixed scalar (or vector) and leave it unchanged.

Q: How do you compute arc length and unit tangent for a vector-valued function r(t)?

Use L = ∫_{a}^{b} |r′(t)| dt, and the unit tangent is r′(t)/|r′(t)| (with the same parameter interval).

Q: Which theorems are listed for relating integrals over curves and regions?

Green’s theorem relates a line integral around C to a double integral over D, and the fundamental theorem for line integrals links ∫_C ∇f · d r to f(r(b)) − f(r(a)).

Oct 13, 2026

Upload by BigDaveOnline

Cheat sheet for FINAL-M252 Calculus III (Fall 2022) compiling essential multivariable calculus results and problem-solving templates. Includes rules for vectors and geometry (norms, dot/cross products), arc length and unit tangent, curvature in 2D and 3D, tangent plane and linear approximation, differentials and chain rule, implicit differentiation, gradient and directional derivatives, and optimization with critical points and the second-derivative (Hessian) test. Also summarizes double/triple integrals (Fubini, polar, cylindrical, spherical), line and surface integrals, work and the fundamental theorem for line integrals, and key theorems such as Green’s, divergence/curl identities, and surface integrals via parametrization.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages2

Reading Time1 min

AccessFree to download

Vector and differential rules

Tangent planes, differentials, chain rule, and implicit differentiation

Gradients and directional derivatives

Optimization and curvature

Max/min tests with Hessian

Curvature in 2D and 3D

Integration formulas

Double and triple integrals (Fubini, polar, cylindrical, spherical)

Line integrals and work

Surface integrals and parametrization

Cheat Sheet for FINAL-M252-Calculus III-Fall 2022 This cheat sheet will be included in the midterm. You do NOT have to memorize these formulas. However, make sure that you understand and know how to apply ALL of them!!! • If a constant is not de􀀌ned (like an α or any other symbol), do NOT assign a value for it. It is a 􀀌xed scalar (or vector) and you should just leave it as it is. • When dealing with functions of several variables, be careful about which variable you are doing the derivative (or integral) with respect to! [Remember the: “Integral (or derivative) with respect to what?”] • If ~r = hx, y, zi then its norm (magnitude) is | ~r| = px2 + y 2 + z 2 ~ • M~ ak~ing a vector unitary: = |~~ ~~|~~ ~~ ||| ~a · ~b = | ~a|| ~b|cos(θ) ~||| ~ · ~b =~ 0 ~⇔ ~a ⊥ ~b ||| proj~a(~b) = ~a|~~ ~~~a·~~ ~~|b . • a × b = |3×3 determinant| (sign − for middle entry!) ||| |a × b| = |a||b|sin(θ) ~ ~ • | ~a × b| = area of parallelogram (= 2×area of triangle) ||| | ~a · (b × ~c) | = volume of parallelepiped.•• TIfa|g(etn)|tectorconst: te=n′′ )runitary: = ~r ′(t)/ | ~r ′(t)| . • Arclength: L = 􀀁tt12 | ~r ′(t)|dt = 􀀁tt12 pf′2 + g′2 + h′2 dt ||| s (t) =􀀁t | ~r ′(τ)| dτ . • Curvature in 3D: κ = 1/radius = 􀀌 dd 􀀌 = 􀀌 ′′ 􀀌 = |~~ ~~~r~~ ~~′|~~ ~~′~~ ~~|~~ ~~~~ ~~′′~~ ~~|~~ ~~ . • Curvature for planar curve y = F (x) in 2D: κ = |F ′′ |/ 􀀀1 + F ′ 2􀀁 3/2 . •• o(tr)al􀀁(bti)(a0l): T(~t)′ | ′~v| (t + ×(t.) = ~v ′(t) = ~r ′′(t) . • Tangential and normal components of the acceleration: ~a = υ′ + κυ 2 = aT + aN ||| υ = | ~r ′ | ||| aT = | ~r ′ · ~r ′′ |/|~r ′ | ||| aN = | ~r ′ × ~r ′′ |/|~r ′ | • Clairant’s theorem: If f , fxy , and fyx are continuous ⇒ fxy = fyx. • Tangent plane to f (x, y) at z0 = f (x0 , y0 ): z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) . ⇒ Linear approximation: z = z0 + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) . ⇒ Di􀀋erentials: dz = fx (x0 , y0 ) dx + fy (x0 , y0 ) dy df dx ∂f dy ∂f • Chain rule (case 1): f = f (x, y) and x = x (t) and y = y (t): = + dt dt ∂x dt ∂y . • Chain rule (case 2): f = f (x, y) and x = x (s, t) and y = y (s, t): df ∂f ∂x ∂f ∂y df ∂f ∂x ∂f ∂y = + and = + ds ∂x ∂s ∂y ∂s dt ∂x ∂t ∂y ∂t dy Fx dz Fx dz Fy • Implicit di􀀋erentiation: F (x, y) = 0: = − ||| F (x, y, z) = 0: = − & = − . dx Fy dx Fz dy Fz •• DirectionGradient:alNdaerblivaative alooperatorn==hah∂,i,: ,(ix, y|||) fxrad(xie,ynt)(af)(xf,(yx), by ,=z)h· ., ∂∂y , ∂∂z if = hfx , fy , fzi. • Direction of steepest ASCENT is given by gradient ∇f(x, y, z) ||| Gradient vector ⊥ level curves. • Tangent plane to F(x, y, z) = k at (x0 , y0 , z0 ): (x−x0 ) Fx +(y−y0 ) Fy +(z−z0 ) Fz = 0 [with partials evaluated at (x0 , y0 , z0 )] . x − x0 y − y0 z − z0 • Normal line to F(x, y, z) = k at (x0 , y0 , z0 ): = = [with partials evaluated at (x0 , y0 , z0 )] . Fx Fy Fz • Remember di􀀋erence between local and global max/min ||| For max/min also check along boundaries! • Max/min will be at critical points: fx = 0 = fy (also check boundaries!) • Second derivative test: fx = 0 = fy and Hessian: D (x, y) = fxx fyy − (fxy)2 = 􀀌 fxxfyx fxyfyy 􀀌 : (a) D > 0 & fxx > 0 ⇒ local min ||| (b) D > 0 & fxx < 0 ⇒ local max ||| (c) D < 0 ⇒ saddle. • Lagrange multiplier: min/max of f with constraint g = k: ∇f(x, y, z) = λ∇g(x, y, z) & g (x, y, z) = k. • Fubini’s theorem: if domain is rectangular ⇒􀀁bx=a 􀀁yd=c f dy dx = 􀀁yd=c 􀀁bx=a f dx dy. • Average: = A~~ ~~(~~1~~D) 􀀃D f (x, y) dA. • Area using double integral: Area(D)= 􀀃D 1 dA. • Type I: 􀀃D f (x, y) dA = 􀀁bx=a 􀀁gy(xg1)(x) f (x, y) dy dx. • Type II: 􀀃D f (x, y) dA = 􀀁yd=c 􀀁xhh()(y) f (x, y) dx dy. • Polar: 􀀃D f (x, y) dA = 􀀃D f (r cos(θ), r sin(θ)) r dr dθ . ||| r2 = x2 + y 2 , x = r cos(θ), y = r sin(θ) . • Cylindrical: 􀀅B f (x, y, z) dV = 􀀅B f (r cos(θ), r sin(θ), z) r dr dθ dz. with r2 = x2 + y 2 + z 2 , x = r cos(θ), y = r sin(θ), z = z. • Spherical: 􀀅B f (x, y, z) dV = 􀀅B f (ρsin(φ)cos(θ),ρsin(φ)

FAQs

What should you do if a constant (e.g., α) is not defined in the problem?

Do not assign a value. Treat it as a fixed scalar (or vector) and leave it unchanged.

How do you compute arc length and unit tangent for a vector-valued function r(t)?

Use L = ∫_{a}^{b} |r′(t)| dt, and the unit tangent is r′(t)/|r′(t)| (with the same parameter interval).

Which theorems are listed for relating integrals over curves and regions?

Green’s theorem relates a line integral around C to a double integral over D, and the fundamental theorem for line integrals links ∫_C ∇f · d r to f(r(b)) − f(r(a)).

Oct 13, 2026

Upload by BigDaveOnline