What is the gradient vector problem for a function with dependent variables?

[evo_vil]

Homework Statement

If [itex]z = f(x,y)[/itex] such that [itex]x = r + t[/itex] and [itex]y = e^{rt}[/itex], then determine [itex]\nabla f(r,t)[/itex]

Homework Equations

[itex]\nabla f(x,y) = <f_x,f_y>[/itex]

The Attempt at a Solution

Now if i follow this the way i think it should be done then i find the partials of f wrt x and y and then simply sub in r and t in the place of x and y respectively...

But if i get del f the normal way i get:

[itex]\nabla f = <f_x+f_y t e^{rt},f_x+f_y r e^{rt}>[/itex]

is this the final/correct answer or am i missing a trick question where i was asked to find del f(r,t) and not del f(x,y)

 

[I like Serena]

Welcome to FP, evo_vil!

Your problem is ambiguous.
If I take it very literal, the answer would be:
[tex]\nabla f(r,t)=<f_x(r,t), f_y(r,t)>[/tex]

However, I can't imagine that this was intended.

I expect that you're supposed to take the gradient from a function f* defined by:
f*(r,t) = f(x(r,t), y(r,t)).
It is not unusual that this function f* is simply called f, although that is ambiguous.

This is what you calculated, and no doubt correct.

 

[evo_vil]

Thanks!

Ive browsed PF for quite a few years, but never participated, so thanks for the welcome...

I think I am just going to go with what I've calculated and see how it goes...
Maybe see if other people get the same thing.

Thanks for your help