Surfaces, Coordinates, and Vector-Valued Functions

Elliptic paraboloid

Surface given by z=(x/a)^2+(y/b)^2.

Hyperbolic paraboloid

Surface given by z=(x/a)^2-(y/b)^2.

Elliptic cylinder

Surface given by 1=(x/a)^2+(y/b)^2.

Hyperbolic cylinder

Surface given by 1=(x/a)^2-(y/b)^2.

Cylindrical coordinates

In (r,θ,z), r: distance from the z-axis, θ: angle in the xy-plane from the +x-axis, z: height above the xy-plane.

Spherical coordinates

In (ρ,θ,ϕ), ρ: distance from the origin, θ: same planar angle as in cylindrical, ϕ: angle down from the positive z-axis.

Cylindrical to rectangular conversion

Convert (2,π/2,5) to (0,2,5).

Spherical to rectangular conversion

Convert (1,π/2,π/2) to (0,1,0).

Differential volume in rectangular

dV = dxdydz.

Differential volume in cylindrical

dV = rdrdθdz.

Differential volume in spherical

dV = ρ^2 sinϕ dρ dϕ dθ.

Differential surface area in rectangular

dA = dxdy, dydz, or dxdz (depending on the plane).

Differential surface area in cylindrical

Lateral: dA = rdθdz; top/bottom disk: dA = rdrdθ.

Differential surface area in spherical

dA = ρ^2 sinϕ dϕ dθ.

Vector-valued function

A function mapping t to ⟨x(t),y(t),z(t)⟩.

Parametrizations of a smooth curve

Infinitely many, via any smooth one-to-one re-parametrization ϕ(u) with ϕ′(u)≠0.

Tangent vector at t0

r′(t0) = ⟨x′(t0),y′(t0),z′(t0)⟩.

Equation of the tangent line at t0

L(s) = r(t0) + s r′(t0).

Arc-length from t=a to t=b

s = ∫(a to b) ∥r′(t)∥ dt.

Speed of a parametric curve at time t

∥r′(t)∥.

Arc-length parametrization

A re-parametrization so that ∥dr/ds∥ = 1.

Velocity vector v(t)

v(t) = r′(t).

Acceleration vector a(t)

a(t) = r′′(t).

Traces of a surface in planes z=constant

Straight lines (for a plane).

Traces of an ellipsoid

Ellipses.

Traces of a one-sheeted hyperboloid

Hyperbolas (vertical) and ellipses (horizontal).

Traces of a two-sheeted hyperboloid

Two separate hyperbola branches (vertical) and ellipses (horizontal) beyond the gap.

Traces of a cone

Straight lines (vertical) and circles (horizontal).

Traces of an elliptic paraboloid

Ellipses (vertical) and parabolas (horizontal).

Traces of a hyperbolic paraboloid

Hyperbolas (vertical) and parabolas opening opposite ways (horizontal).

Contour map

A drawing of level curves {(x,y)∣f(x,y)=k} labeled by k.

Level curve

The set {(x,y)∣f(x,y)=k} for fixed k.

Level curves of f(x,y)=3+x^2+y^2

Concentric circles centered at the origin.

Testing a multivariable limit

Evaluate along different paths; if any two give different values, the limit does not exist.

Limit DNE

Exhibit two paths to (a,b) with different limit values.

Partial Derivative fx

fx(a,b)=lim h→0 (f(a+h,b)−f(a,b))/h

Partial Derivative fy

fy(a,b)=lim k→0 (f(a,b+k)−f(a,b))/k

Clairaut's Theorem

If fxy and fyx are continuous near (a,b), then fxy(a,b)=fyx(a,b).

Tangent Plane

z=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b).

Linearization L(x,y)

L(x,y)=f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b).

Gradient ∇f

∇f=⟨fx,fy⟩; points in direction of steepest increase, magnitude = max rate.

Gradient and Level Curves

It is perpendicular to level curves.

Directional Derivative Du f

The rate of change of f in direction of unit vector u.

Directional Derivative Formula

Du f=∇f⋅u.

Directional Derivative in terms of θ

Du f=∥∇f∥cosθ.