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

Question

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 1

Gradient of a vector field (or a multi-valued function $f: R^m\to R^n$) is jacobian of the multi-valued function $f$, where each row $r_i$ of the $\text{Jacobian}(f)$ represents the gradient of $f_i$ (remember, each component $f_i$ of the multi-valued function $f$ is a scalar).

Answer 2

The gradient of a vector field corresponds to finding a matrix (or a dyadic product) which controls how the vector field changes as we move from point to another in the input plane.

Details: Let $ \vec{F(p)} = F^i e_i = \begin{bmatrix} F^1 \\ F^2 \\ F^3 \end{bmatrix}$ be our vector field dependent on what point of space we take, if step from a point $p$ in the direction $ \epsilon \vec{v}$, we have:

$$ \vec{F(p+ \epsilon \vec{v})} = F^i(p+ \epsilon v) e_i= F^i(p) e_i + \epsilon \left(v \cdot \nabla F^i(p) \right) e_i$$

But, what is $( v \cdot \nabla F^i ) e_i= \begin{bmatrix} \nabla F^1 \\ \nabla F^2 \\ \nabla F^3 \end{bmatrix} v= \begin{bmatrix} \partial_1 F^1 & \partial_2 F^1 & \partial_3 F^1 \\ \partial_1 F^2 & \partial_2 F^2 & \partial_3 F^2 \\ \partial_1 F^3 & \partial_2 F^3 & \partial_3 F^3 \end{bmatrix} \begin{bmatrix} v^1 \\ v^2 \\ v^3 \end{bmatrix}$

And that's what the gradient of a vector field is, a big matrix controls how the gradient vector changes when we move in any direction of the input space.

Note that I have assumed the eulicdean basis (i.e: i,j,k)