Test 3

Cylindrical Coordinates

1. Z-Simple

2. R: xy-plane

Spherical Coordinates

(p,θ,ϕ)

(spherical coordinates) x =

psinθcosϕ

(spherical coordinates) y =

psinθsinϕ

(spherical coordinates) z =

pcosϕ

(spherical coordinates) r =

psinϕ

x² + y² + z² =

p²

(spherical coordinates) p =

√[x² + y² + z²]

(spherical coordinates) cosϕ =

z / p

(spherical coordinates) tanθ =

y / x

x² + y² =

ρ²sin²(ϕ)

√[x² + y²] =

ρsin(ϕ)

Right Circular Cone Formula

z² = x² + y²

Cone with a Constant Slope Formula

z² = x² + y² / tan²θ

Half Plane Formula

Ax + By + Cz ≥ D OR Ax + By + Cz ≤ D

Circular Cylinder Formula

x² + y² = r²

ϕ=0

Point lies on the z-axis (straight up)

ϕ=π/2

Point lies in the xy-plane

ϕ=π

Point lies on the negative z-axis (straight down)

ϕ

The angle between a point and the positive z-axis in spherical coordinates.

• It measures how far "down" from the z-axis a point lies, ranging from:

• 0 ≤ ϕ ≤ π​

The field is not conservative because the paths have the same terminal and initial points, yet different line integrals.

Area of Elliptic Cylinder

A_base​ = π(a)(b)

(Trig Sub) √[1-x²]

x = sinθ

(Trig Sub) √[1+x²]

x = tanθ

(Trig Sub) √[x²-1]

x = secθ

1 - sin²θ =

cos²θ

1 + tan²θ =

sec²θ

sec²θ − 1 =

tan²θ

∫₀^2π |sinx| dx =

∫₀^π sinx dx + ∫_π^2π -sinx dx

ϕ = 0

Pointing straight up along the positive z-axis

ϕ = π/2​

Pointing flat in the xy-plane

ϕ = π

Pointing straight down along the negative z-axis

Integral can be separated

The integrand is a product of single-variable functions

• e.g., f(x)g(y)h(z)

The region of integration is a rectangular box

• e.g., [a,b]×[c,d]×[e,f]

Integrals cannot be separated

The variables are entangled in the integrand

• e.g., x+y, xy, x²+z²

Any bound of one variable depends on another variable

• e.g., y goes from 0 to x, or polar regions like r to θ

Semicircle Formula

y = √[r² - x²]

c • ∫ √[r² - x²]

½A = ½πr² • c