Math 119- Week 6

Cylindrical coordinates =

(r, θ, z)

where:

• r, θ are polar coordinates

• z is the height

convert to (x, y, z):

• x = r cos θ

• y = r sin θ

• z = z

r = √(x² + y²)

where r ≥ 0 and θ ∈ [0, 2π].

Jacobian of transformation for cylindrical coords =

dxdydz = rdrdθdz

Spherical coordinates =

(r, φ, θ)

where:

• r = distance from origin

• θ = angle with x-axis // “azimuthal” angle

• φ = angle with z-axis // “zenith” angle

convert to (x, y, z)

• x = rsinφcosθ

• y = rsinφsinθ

• z = rcosφ

r = √(x² + y² + z²)

where r ≥ 0, 0 ≤ θ ≤ 2π, and 0 ≤ φ ≤ π.

azimuthal angle meaning =

Spherical coordinates:

• angle with x-axis

• θ

zenith angle meaning =

Spherical coordinates:

• angle with z-axis

• φ

Jacobian of transformation for spherical coordinates =

dxdydz = r²sinφdrdφdθ

When can double/triple integral be split into product of single integrals =

• bounds are constant

• entire integrand can be rewritten as a product of functions which all depend on one variable

how to solve improper integral =

let a → ∞

• set lim_{a→∞ c}∫^a f(x)

• evaluate integral with L’Hopitals rule if necessary

when can L’Hopitals rule be used

if limit is indeterminant

• ∞/∞

• 0/0