Question concerning curl for finding a conservative force field

[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 don't 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

 

[GreyBadger]

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.

 

[sleventh]

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