Hello everybody! My question has to do with the parameterization of 3D surfaces from 3 variables to 2. Specifically, I'm trying to figure out an aspect of the cross product of the directional derivatives of the parameterization to solve flux integrals. Trying to convert: ∫∫SF dS = ∫∫DF(r(u,v)) (ru x rv) du dv I hope that this is clear. 1. The problem statement, all variables and given/known data What is the difference between r(R, theta) and r(theta, R), and its effects on the cross product: rR X rθ Suppose you have a circle on the x-y plane with: X = R * cos(θ) Y = R * sin(θ) Z = 0 Where 0 ≤ R ≤ 1, 0 ≤ θ ≤ 2π Now, this can be written as r(R, theta) or r(theta, R), but then does r still end up being the same?: X = R * cos(θ) Y = R * sin(θ) Z = 0 I know that: rR X rθ = - rθ X rR This cross product determines the orientation of the surface, so this is important to understand. 2. Relevant questions How do you determine which variable gets to be the radius and which gets to be the angle? How do you determine which directional derivative gets to go first in the cross product? This is especially relevant when the variables that I'm converting to are U and V instead of something polar. 3. The attempt at an explanation My guess would be that it would be something to do with the right-hand rule, where angles always go CCW. But then this is not easy to visualize, nor is it a really good explanation. EDITED: Not magnitude of cross product.