Final Exam

Vector-valued function

A function r(t)=⟨x(t), y(t), z(t)⟩ that assigns a vector in ℝ² or ℝ³ to each real t

Velocity vector v(t)

r′(t)=⟨x′(t), y′(t), z′(t)⟩

Speed

‖v(t)‖=√[x′(t)²+y′(t)²+z′(t)²]

Acceleration vector a(t)

v′(t)=r″(t)

Arc length formula

L=∫ₐᵇ‖r′(t)‖ dt

Unit tangent vector T(t)

r′(t)/‖r′(t)‖

Unit normal vector N(t)

T′(t)/‖T′(t)‖

Curvature κ

‖r′(t)×r″(t)‖/‖r′(t)‖³

ds/dt vs. ds relationship

d/ds = (1/‖r′(t)‖) d/dt

Tangential accel. a_T

d/dt‖r′(t)‖

Normal accel. a_N

κ‖r′(t)‖²

Vector projection proj_v u

(u·v)/(v·v)·v

Work along a curve

W=∫_C F·dr = ∫ₐᵇ F(r(t))·r′(t) dt

Conservative vector field

F is conservative ⇔ ∇×F=0 on a simply connected domain

Fundamental Theorem of Line Integrals

∫_C F·dr = φ(r(b))−φ(r(a)) if F=∇φ

General 2nd-degree in x,y

Ax²+Bxy+Cy²+Dx+Ey+F=0

Circle criteria

B=0, A=C ⇒ (x−h)²+(y−k)²=r²

Ellipse standard form

(x−h)²/a²+(y−k)²/b²=1, a>b

Hyperbola standard form

(x−h)²/a²−(y−k)²/b²=1

Parabola standard form

(y−k)²=4p(x−h)

Eccentricity values

circle e=0; ellipse 0

Focus-directrix definition

dist(P,F)/dist(P,directrix)=e

Rotate axes to remove xy

K: cot(2θ)=(A−C)/B

Center a conic

Complete the square in x,y and shift

Degenerate conics

B²−4AC=0 ⇒ parabola or pair of lines/point

Rectangular→Polar

r=√(x²+y²), θ=atan2(y,x)

Polar→Rectangular

x=r cosθ, y=r sinθ

Area in polar

A=∫ₐᵇ∫₀^R(θ) r dr dθ

Polar arc length

L=∫ₐᵇ√[r(θ)²+r′(θ)²] dθ

Smooth parametric curve

x′(t),y′(t) continuous and never zero

Parametric surface area

A=∬D‖ru×r_v‖ du dv

Partial derivative fx(a,b)

lim_{h→0}[f(a+h,b)−f(a,b)]/h

Gradient ∇f

⟨fx, fy⟩

Directional derivative

D_u f(a,b)=∇f·u

Tangent plane

z−f(a,b)=fx(a,b)(x−a)+fy(a,b)(y−b)

Differentiability criterion

Error of linear approximation is o(√(h²+k²))

Chain rule (2-vars)

dz/dt = fx dx/dt + fy dy/dt

Ellipsoid

x²/a² + y²/b² + z²/c² = 1

Cylinder along z

x²/a² + y²/b² = 1

Elliptic paraboloid

z = x²/a² + y²/b²

Hyperbolic paraboloid

z = x²/a² − y²/b²

One-sheet hyperboloid

x²/a² + y²/b² − z²/c² = 1

Two-sheet hyperboloid

−x²/a² − y²/b² + z²/c² = 1

Identify quadric

Match signs of squared terms to standard forms

Double integral volume

V=∬_D f(x,y) dA

Polar element in 2D

dA = r dr dθ

Triple integral (cyl)

dV = r dr dθ dz

Spherical coords

x=ρ sinφ cosθ; y=ρ sinφ sinθ; z=ρ cosφ

Spherical volume

element dV=ρ² sinφ dρ dφ dθ

Choose coords by symmetry

Circle→polar; cylinder→cylindrical; sphere→spherical

Green’s Theorem

∮C P dx+Q dy = ∬D(∂Q/∂x−∂P/∂y)dA

Divergence Theorem

∬S F·n dS = ∭E (∇·F) dV

Stokes’ Theorem

∮C F·dr = ∬S (∇×F)·n dS

Orientation rule

Use right-hand rule for boundary vs normal

Green’s area trick

If P=−y/2, Q=x/2 ⇒ ∮x dy−y dx=2∬dA

Dot product

u·v = u₁v₁+u₂v₂+u₃v₃

Perp test

u·v=0 ⇒ u⊥v

Cross product

u×v = det[[i,j,k],[u₁,u₂,u₃],[v₁,v₂,v₃]]

Area via cross

‖u×v‖ = area of parallelogram

Line eqn

r(t)=P₀ + t d

Symmetric line

(x−x₀)/d₁=(y−y₀)/d₂=(z−z₀)/d₃

Plane eqn

A(x−x₀)+B(y−y₀)+C(z−z₀)=0

Point-plane distance

|Ax₁+By₁+Cz₁+D|/√(A²+B²+C²)

Multivar limit def

∀ε>0 ∃δ>0 s.t.‖(x,y)−(a,b)‖<δ⇒|f(x,y)−L|<ε

Path test for DNE

Different paths (e.g., y=mx, x=0) give different limits

Continuity at (a,b)

lim_(x,y)->(a,b)f(x,y)=f(a,b)

Polar path test

Convert to (r,θ); if limit depends on θ, no limit

Linearization L(x,y)

f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)

Approximate with L

f(a+Δx,b+Δy)≈L(a+Δx,b+Δy)

Total differential dz

dz = fx dx + fy dy

First step problem-solving

Read carefully: givens, asks, constraints

Choosing integral method

Rational→partial fractions; trig→identities; composite→u-sub

Choosing coords

By symmetry: circle→polar; cylinder→cylindrical; sphere→spherical

Using Green/Stokes/Divergence

✔ differentiability; ✔ orientation; compute interior integral

Vector surface integral steps

Parametrize r(u,v); compute ru×rv; integrate F·(ru×rv)

Line integral steps

Parametrize (x(t),y(t)), get dx,dy, substitute, integrate

Domain check

Identify endpoints and points where integrand undefined

Antiderivative check

Differentiate candidate to recover integrand

Orientation confirm

Use right-hand rule for curve/surface

Multivar limit test

Approach along multiple paths

Partial derivative check

Hold one var constant; compare mixed partials fxy & fyx

Gradient normal check

Dot gradient/normal with tangent = 0

Bounds check

Sketch region; project to find correct limits

Parametrization check

Check derivative ≠0 and covering exactly once

Conservative 2D test

∂P/∂y = ∂Q/∂x on simply connected domain