1. ## Flow Lines/Curl Question?

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.

2. 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.

$\displaystyle \displaystyle \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ F_x & F_y & F_z \end{vmatrix}=\left(\frac{\partial z}{\partial y}-\frac{\partial y}{\partial z}\right)\mathbf{i}+\left(\frac{\partial x}{\partial z}-\frac{\partial z}{\partial x}\right)\mathbf{j}+\left(\frac{\partial y}{\partial x}-\frac{\partial x}{\partial y}\right)\mathbf{k}$

3. there was a second part that said what can you conclude about the effect on local circulation of grad f?

I'm not sure how to find the mixed partial derivatives though

4. The ones I posted?

5. yea because it never actually mentions any function
it just says "a twice differentiable function"

6. If you take the determinant of the matrix (cross product), you will obtain those partial derivatives.