What does it mean to take the gradient of a vector field?

What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field.

Answer

The gradient of a vector is a tensor which tells us how the vector field changes in any direction. We can represent the gradient of a vector by a matrix of its components with respect to a basis. The $(\nabla V)_{\text{ij}}$ component tells us the change of the $V_j$ component in the $\pmb{e}_i$ direction (maybe I have that backwards). You can check out the Wikipedia article for the details of calculating the components.

To get a physical picture of its meaning we can decompose it into
1) the trace (the divergence)
2) an anti-symmetric tensor (the curl)
3) a traceless symmetric tensor (the shear)

If the vector field represents the flow of material, then we can examine a small cube of material about a point.
The divergence describes how the cube changes volume.
The curl describes the shape and volume preserving rotation of the fluid.
The shear describes the volume-preserving deformation.

Attribution
Source : Link , Question Author : fred , Answer Author : Timothy Wofford

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