The document you uploaded contains several physics and mathematics problems. Here is a list of all the questions:

1. Calculate the Rot of the following vectors: a. $\mathbf{A} = x\hat{i} + y\hat{j} + z\hat{k}$ b. $\mathbf{B} = r \hat{i} \cdot \cos(\phi) + 2 \hat{j}$ c. $\mathbf{C} = r \hat{e}_\theta - 2z \hat{e}_\phi$

2. Calculate the gradient (grad) of the following functions: a. $A = 10x^2 + 12y^2 + 25z^2$ b. $B = \sin(r)\cos(\phi)$ c. $C = 2\cos(\theta)\sin(\phi)$

3. Find the divergence at point P of the following vectors: a. $\mathbf{A} = [\sin(\theta) \cos(\phi), \sin(\theta)\cos(\phi), 2]$ b. $\mathbf{B} = [0, 10.5, 5]$ c. $\mathbf{C} = [z, 5, 2]$

4. Calculate the divergence and curl of the following vectors: a. $\mathbf{A} = (2x^2 + y^2) \hat{i} + (3z + y^2) \hat{j} + x^2 \hat{k}$ b. $\mathbf{B} = (r \sin(\phi)) \hat{i} + (r \cos(\phi)) \hat{j}$ c. $\mathbf{C} = (sin(\theta)\cos(\phi)) \hat{i} + (cos(\theta)\sin(\phi)) \hat{j}$

5. Consider a conductor with a cylindrical shape. Given the frequency and amplitude, calculate the current density and electric field inside the conductor.

6. Consider a wire with a uniform electric field. Calculate the Poynting vector and the power input into the wire.

7. Find the charge distribution causing an electric field, given specific conditions for $\mathbf{D}$.

8. In a region with $\epsilon_r = 2$, calculate the charge distribution generated by a specific electric field configuration.

9. For a time-varying electric and magnetic field, solve for the electric field and check if Maxwell's equations are satisfied.

10. For a time-varying electric field, solve for the magnetic field and calculate the induced current in a loop.

If you need help with any specific question or would like to solve them step-by-step, let me know! Would you like to go through these one by one, or is there a specific question you need help with?