Gradient of Vector A: What Does It Mean?

[enricfemi]

$$\nabla$$$$\stackrel{\rightarrow}{A}$$

when a gradient operater act on a vector,what is it stand for ?

 

[D H]

$$\nabla$$$$\stackrel{\rightarrow}{A}$$

when a gradient operater act on a vector,what is it stand for ?

Visually, what you wrote looks like

$$\nabla_{\vec A}$$

The title of the thread and your LaTeX suggests you meant

$$\nabla \vec A$$

These are two different things. The first is an operator, the gradient with respect to the components of $\vec A$, rather than the normal gradient which is take with respect to spatial components. The second form is the gradient of a vector. It is a second-order tensor. If $$\vec A = \sum_k a_k \hat x_k$$,

$$(\nabla \vec A)_{i,j} = \frac{\partial a_i}{\partial x_j}$$

BTW, it is best not to separate things the way you did in the original post. Here is your original equation as-is:

$$\nabla$$$$\stackrel{\rightarrow}{A}$$

Now look at how this appears when written as a single LaTeX equation:

$$\nabla\stackrel{\rightarrow}{A}$$

 

[dakg]

The second form is the gradient of a vector. It is a second-order tensor. If $$\vec A = \sum_k a_k \hat x_k$$,

$$(\nabla \vec A)_{i,j} = \frac{\partial a_i}{\partial x_j}$$

Does this make a matrix using row i and column j for the entries?

 

[D H]

Yes.

 

[dakg]

Thank you.