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.
 A: Gradient of a vector field is intuitively the Flux/volume leaving out of the differential volume dV. Visualise  in 2D first. Suppose you have a vector field E in 2D. Now if you plot the Field lines of E and take a particular Area (small area..), Divergence of E is the net field lines, that is, (field line coming out of the area minus field lines going into the area).
Similarly in 3D, Divergence is a measure of (field lines going out - field lines coming in). If you mathematically implement this you see you get 3 terms of partial derivatives added, which essentially adds the total net field lines.
For a scalar field(say F(x,y,z) ) it represents the rate of change of F along the the 3 perpendicular ( also called orthonormal ) vectors you defined your system with (say x, y, z ). 
A: 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.
A: 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)
A: It depends on how you define the gradient operator.  In geometric calculus, we have the identity $\nabla A = \nabla \cdot A + \nabla \wedge A$, where $A$ is a multivector field.  A vector field is a specific type of multivector field, so this same formula works for $\vec v(x,y,z)$ as well.
So we get $\nabla\vec v = \nabla \cdot \vec v + \nabla \wedge \vec v$.  The first term should be familiar to you -- it's just the regular old divergence.  However the second term is a different type of object entirely (actually, it's a generalization of the familiar $3$D curl $\nabla \times \vec u$ that works in any dimension).
In the same way that a vector field can be though of as associating with every point in your domain an oriented line segment (a vector), $\nabla \wedge \vec v$ associates with every point in your domain an oriented plane segment (which we call bivectors).  So $\nabla \wedge \vec v$ is called a bivector field.
So to answer your question, the gradient of a vector field is the sum of a scalar field and a bivector field.
A: Assume the vector $\vec{\bf F} = (F_1, F_2, F_3)$ exists in a 3D space with basis $x_1, x_2, x_3$, then its gradient is the 3 × 3 matrix: $\partial_i$$F_j$
$\nabla\vec{\bf F}=\left( {\begin{array}{c}
\frac{\partial F_1}{\partial x_1}&\frac{\partial F_1}{\partial x_2}&\frac{\partial F_1}{\partial x_3}\\
\frac{\partial F_2}{\partial x_1}&\frac{\partial F_2}{\partial x_2}&\frac{\partial F_2}{\partial x_3}\\
\frac{\partial F_3}{\partial x_1}&\frac{\partial F_3}{\partial x_2}&\frac{\partial F_3}{\partial x_3}\\
\end{array}} \right).$
That is, each column is a "usual" gradient of the corresponding scalar component function.
A: 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).
