The curl of the gradient of a twice continuously differentiable function R3 -> R is identically zero.
Prove this by direct computation of the required mixed partial derivatives.
The curl of the gradient of a twice continuously differentiable function R3 -> R is identically zero.
Prove this by direct computation of the required mixed partial derivatives.
I am not 100% on what you are asking. Is there more to this question? However, I have devised a guess which is provided below.
$\displaystyle \displaystyle \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k}\\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\
F_x & F_y & F_z
\end{vmatrix}=\left(\frac{\partial z}{\partial y}-\frac{\partial y}{\partial z}\right)\mathbf{i}+\left(\frac{\partial x}{\partial z}-\frac{\partial z}{\partial x}\right)\mathbf{j}+\left(\frac{\partial y}{\partial x}-\frac{\partial x}{\partial y}\right)\mathbf{k}$