Derivatives of parametric functions (or whatevs)

[variety]

Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused. Can you guys help me out?!?

 

[tiny-tim]

hi variety!

Suppose I have functions x(t,s) and y(t,s) … and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused.

does it make any sense?

suppose x is height, and t and s are north and south …

then x is a hill, and there are contour lines along which x does not change, but y does change …

what could be the meaning of dy/dx ? ​