Partial derivatives - verify solution?

[jjou]

[SOLVED] partial derivatives - verify solution?

Let $$f:\mathbb{R}^3\rightarrow\mathbb{R}$$, $$g:\mathbb{R}^2\rightarrow\mathbb{R}$$, and $$F:\mathbb{R}^2\rightarrow\mathbb{R}$$ be given by
$$F(x,y)=f(x,y,g(x,y))$$.
1. Find DF in terms of the partial derivatives of f and g.
2. If F(x,y)=0 for all (x,y), find $$D_1g$$ and $$D_2g$$ in terms of the partial derivatives of f.

My solution:
1. $$DF=D_1F+D_2F=(f_1+f_3g_1)+(f_2+f_3g_2)$$
2. If $$f_3\neq0$$, then we have the partials of F being zero, so:
$$g_1=-f_1/f_3$$ and $$g_2=-f_2/f_3$$. However, if $$f_3=0$$ then we have $$f_1=f_2=0$$.

My concern is with the last part of 2. If $$f_3=0$$, then I cannot make any statement about the partials of g. Am I doing something wrong?


NOTE: $$f_1$$ refers to differentiation of f by the first variable.

 

[HallsofIvy]

No, that's completely true. If F is, in fact, NOT a function of g, then no information about F can tell you anything about g!

 

[jjou]

Thanks! :)