What's the physical meaning of Curl of Curl of a Vector Field?

  • #1
PLAGUE
16
0
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 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?
 
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  • #2
PLAGUE said:
How does gradient of divergence and divergence of gradient relate to curl?
This might help:

 
  • #3
What those terms mean physically can only be answered in the context of a physical situation. If you work enough in physics, you will come across physical situations that need those terms. That will answer your question
 
  • #4
Meir Achuz said:
What those terms mean physically can only be answered in the context of a physical situation. If you work enough in physics, you will come across physical situations that need those terms. That will answer your question
How about
$$\nabla \times (\nabla \times \mathbf{B}) = \nabla (\nabla \cdot \mathbf{B}) - \nabla^2 \mathbf{B}$$?

Here B is the magnetic field. What do they represent here physically?
 
Last edited:
  • #6
PLAGUE said:
TL;DR Summary: How does gradient of divergence and divergence of gradient relate to curl?

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, ∇(∇⋅A) is a vector valued function (since it is gradient of divergence). How can we add these Vector function and scaler function?
There are two Laplace operators, the scalar Laplacian and the vector Laplacian. The Laplacian ## \nabla^2 ## used in the equation $$ \nabla \times (\nabla \times \mathbf A) = \nabla (\nabla \cdot \mathbf A) - \nabla^2 \mathbf A $$ is the vector Laplace operator. (https://en.wikipedia.org/wiki/Laplace_operator#Vector_Laplacian) The vector Laplacian is sometimes represented by a regular hexagram to distinguish the vector Laplace operator from the scalar Laplace operator.

The physical meaning of the vector Laplacian is the same as the physical meaning of the scalar Laplacian. The Laplacian of a scalar field at point p measures the amount by which the average of the scalar over small balls centered at p differs from the scalar at p while the Laplacian of a vector field at point p measures the amount by which the average of the vector over small balls centered at p differs from the vector at p.
 

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