The uploaded image contains mathematical expressions related to the gradient, divergence, and curl of a vector, as well as the use of the ∇ (nabla) operator in Cartesian coordinates.

The document presents the following key points:

1. Gradient of a function: $\nabla f = \hat{i} \frac{\partial f}{\partial x} + \hat{j} \frac{\partial f}{\partial y} + \hat{k} \frac{\partial f}{\partial z}$

2. Divergence of a vector $A$: $\nabla \cdot \mathbf{A} = \hat{i} \frac{\partial A_x}{\partial x} + \hat{j} \frac{\partial A_y}{\partial y} + \hat{k} \frac{\partial A_z}{\partial z}$

3. Curl of a vector $A$: $\nabla \times \mathbf{A} = \hat{i} \frac{\partial A_x}{\partial x} + \hat{j} \frac{\partial A_y}{\partial y} + \hat{k} \frac{\partial A_z}{\partial z}$

The document also lists several identities involving the ∇ operator that need to be proven, such as:

• $\nabla \cdot \mathbf{r} = 3$
• $\nabla \times \mathbf{r} = 0$
• $\nabla r = \hat{r}$
• $\nabla \hat{r} = \frac{2}{r}$
• $\nabla f(r) = \hat{r} \frac{\partial f}{\partial r}$
• $\nabla \left( \frac{1}{r} \right) = -\hat{r} \frac{1}{r^2}$
• $\nabla \cdot \left( \frac{\mathbf{r}}{r^3} \right) = 0$ if $r \neq 0$

Would you like me to explain any specific identity or concept from this material in more detail? Or perhaps assist with the proofs of these identities?