- #1

PLAGUE

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- TL;DR Summary
- How does gradient of divergence and divergence of gradient relate to curl?

So, curl of curl of a vector field is, $$\nabla \times (\nabla \times \mathbf{A}) = \nabla (\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$$

Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise.

The laplacian is divergence of the gradient of a scalar function. The Laplacian takes a scalar valued function and gives back a scalar valued function. While, $$\nabla (\nabla \cdot \mathbf{A})$$ is a vector valued function (since it is gradient of divergence). How can we add these Vector function and scaler function?

And

And

Now, curl means how much a vector field rotates counterclockwise. Then, curl of curl should mean how much the curl rotate counterclockwise.

The laplacian is divergence of the gradient of a scalar function. The Laplacian takes a scalar valued function and gives back a scalar valued function. While, $$\nabla (\nabla \cdot \mathbf{A})$$ is a vector valued function (since it is gradient of divergence). How can we add these Vector function and scaler function?

And

**how does $$\nabla (\nabla \cdot \mathbf{A})$$ and $$\nabla^2 \mathbf{A}$$, i.e. gradient of divergence and divergence of gradient relate to curl?**And

**what does $$\nabla (\nabla \cdot \mathbf{A})$$ and $$\nabla^2 \mathbf{A}$$ mean physically?**