Solving Vector Area dS of Sphere: Find Mistake!

[aaaa202]

Homework Statement

I wonna calculate the vector area dS of a sphere, but for some reason my result gets mixed areound. I need a trained eye to see where I make a silly mistake.

Homework Equations

Parametrization of a sphere:

(x,y,z) = r(cos$\phi$sin$\theta$,sin$\phi$cos$\theta$,cosθ)

The Attempt at a Solution

So ∂r/∂$\phi$ = r(cosθcos$\phi$,-sinθsin$\phi$,sinθ)
and
∂r/∂θ = r(-sin$\phi$sinθ,cos$\phi$sinθ,0)

and dS = ∂r/∂$\phi$ x ∂r/∂θ = r² (sinθcosθcos$\phi$,sin$\phi$sinθ², ...) dθd$\phi$

But the first two terms should be switched around according to my notes! Where do I go wrong? :(

 

[LCKurtz]

Homework Statement

I wonna calculate the vector area dS of a sphere, but for some reason my result gets mixed areound. I need a trained eye to see where I make a silly mistake.

Homework Equations

Parametrization of a sphere:

(x,y,z) = r(cos$\phi$sin$\theta$,sin$\phi$cos$\theta$,cosθ)

The Attempt at a Solution

So ∂r/∂$\phi$ = r(cosθcos$\phi$,-sinθsin$\phi$,sinθ)
and
∂r/∂θ = r(-sin$\phi$sinθ,cos$\phi$sinθ,0)

and dS = ∂r/∂$\phi$ x ∂r/∂θ = r² (sinθcosθcos$\phi$,sin$\phi$sinθ², ...) dθd$\phi$

But the first two terms should be switched around according to my notes! Where do I go wrong? :(

Well, you have the $\theta$ and $\phi$ reversed from the usual math notation for spherical coordinates, but I guess some physicists do that. But when you differentiated with respect to $\phi$ it looks like you differentiated the $\theta$ variables, and conversely.

 

[aaaa202]

nvm.. Found my mistake - the parametric equation was wrong.