*bump*, please help
After literally hours of "no, and what and how", I need help with the following task:
f(u,v)= g(x(u,v),y(u,v))
x(u,v)= u^2- v^2
y(u,v)= 2uv
Expressing f'u and f'v as partial derivatives of g I get:
f'u= (dg/dx)*(dx/du) + (dg/dy)*(dy/du)
f'v= (dg/dx)*(dx/dv) + (dg/dy)*(dy/dv)
(written in crooked d's)
Now I'm supposed to find all functions f that satisfy u*f'u - v*f'v=0
The hardest part (which confuse me the most because there are so many steps involving the chain rule is the following): "Express f''uu+ f''vv through partial derivatives of g. Simplify as much as possible"
Please help me with either of the tasks, I need this by monday and I'm so confused and tired of staring at this task!
So you know that . You can write as , and as (to save writing). You can also find and (from the formulas for x(u,v) and y(u,v)). That has the effect of writing as
.
Now do the same for the other partial derivative .
Having done that, multiply the formula for by u, multiply the formula for by v, and subtract. That will tell you something about the possible functions that satisfy .
Differentiate the formula (*) partially with respect to u, using the chain rule and the product rule: . Now you need to figure out what and are. To do that, use the formula (*), but replace by . That tells you that (and you can get a similar expression for ).
That way, you get an expression for in terms of the second partial derivatives of g. Do the same for , add, and simplify ... .