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.