# Question concerning curl for finding a conservative force field

1. Sep 28, 2009

### sleventh

Hello all,
I understand the fact that the principles
LaTeX Code: F= \\nabla \\phi .

LaTeX Code: \\nabla \\times F = 0 .

must apply in order for a force field to be conservative however what i dont get is why showing:

LaTeX Code: f_y= g_x, f_z= h_x, g_z= h_y
where subscripts are what you are taking the derivative with respect to.

is a means to prove the above laws are in effect. i assume it has to do with the fact that subtracting the partial derivatives will give you zero. thank you very much for any help

sleventh

2. Sep 29, 2009

You should specify what $$f$$, $$g$$ and $$h$$ are. It does look like you're using the fact that you can jointly express the two statements as

$$\nabla\times\nabla\phi = \mathrm{det}\begin{pmatrix}\hat{\mathbf{i}} && \hat{\mathbf{j}} && \hat{\mathbf{k}}\\\delta_x && \delta_y && \delta_z\\ \delta_x\phi && \delta_y\phi && \delta_z\phi\end{pmatrix}$$,
meaning, for example,
$$\hat{\mathbf{i}}(\delta_y\delta_z - \delta_z\delta_y)\phi=0,$$
but not knowing $$f$$, $$g$$ and $$h$$, I can't comment further.

3. Sep 29, 2009

### sleventh

oh haha that should have been obvious thank you very much that clears everything up perfectly