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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.

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    and in the special case of the vector being the electromagnetic vector field $\vec A$, is $\vec{\nabla} \vec{A} =0$?2018-03-21

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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.

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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).

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    Hope this edit is acceptable.2013-03-31
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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.

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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$

$\nabla\vec{\bf F}=\left( {\begin{array}{c} \frac{\partial F_1}{\partial x_1}&\frac{\partial F_2}{\partial x_1}&\frac{\partial F_3}{\partial x_1}\\ \frac{\partial F_1}{\partial x_2}&\frac{\partial F_2}{\partial x_2}&\frac{\partial F_3}{\partial x_2}\\ \frac{\partial F_1}{\partial x_3}&\frac{\partial F_2}{\partial x_3}&\frac{\partial F_3}{\partial x_3}\\ \end{array}} \right).$

That is, each column is a "usual" gradient of the corresponding scalar component function.

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    I think that's wrong because the correct result is the transpose of what you wrote. take a look at the page 7 of [this document](http://web.iitd.ac.in/~pmvs/courses/mcl702/notation.pdf):2018-04-20
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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 ).