It looks like you've uploaded an image containing a mathematical expression. The equation shown in the image is:

$\text{curl}(\nabla f(\mathbf{r})) = 0$

This represents the condition that the curl of the gradient of a scalar function $f(\mathbf{r})$ is zero.

Explanation:

• The gradient $\nabla f(\mathbf{r})$ is a vector field that points in the direction of the greatest rate of change of the scalar function $f$.
• The curl of a vector field measures the rotation or circulation of the field around a point.
• For any scalar field $f$, the curl of its gradient is always zero. This is a fundamental property in vector calculus, often written as: $\text{curl}(\nabla f) = 0$

This implies that the gradient of a scalar field is irrotational (there's no "twisting" around any point).

Would you like more details on this topic or perhaps a deeper explanation about how this result is derived?

Here are some related questions you might find interesting:

1. What does it mean for a vector field to be irrotational?
2. How is the gradient of a scalar field related to the potential of a vector field?
3. Can you show an example of calculating the curl of the gradient of a scalar function?
4. What are the conditions for a vector field to be conservative?
5. How does the divergence theorem relate to the curl of a vector field?

Tip: Remember that the curl operation measures the "rotational" behavior of a vector field, while the gradient measures the rate of change of a scalar field. The two operations are fundamental to understanding the behavior of fields in physics and engineering.