Multivariable calculus lecture 13

Surface area integral for graph surfaces like z=g(x,y)

3 formulas (depending on how u define variables for surface)

remove “f” —> SA

Surface integral of a scalar function using parametrized surfaces, where r is defined w/ u and v

2 ways to parametrize surfaces

• if one variable is defined with other two variables (eg x=√(y²+z²)): make the other two variables __ and _

• if not (eg. x²=y²+z²), parametrize _____

Standard Parametrizations

• Sphere:
  r(ϕ,θ)=⟨psin⁡ϕcos⁡θ,  psin⁡ϕsin⁡θ,  pcos⁡ϕ⟩
  dS=_____

• Cylinder:
  r(θ,z)=⟨rcos⁡θ,  rsin⁡θ,  z⟩
  dS=____

u, v

one side

p²sin⁡ϕ dϕ dθ

r dθ dz

Surface integral of vector field/flux integral for graph surfaces like z=g(x,y)

and for parametrically defined surfaces

Flux if S is a simple closed surface (like sphere): divergence theorem