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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... thank you.

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Did you have a look at this? – draks ... Jun 11 '12 at 8:42
The short answer is: the gradient of the vector field $\sum v_i(x, y, z)e_i$, where $e_i$ is an orthonormal basis of $\mathbb{R}^3$, is the matrix $(\partial_i v_j)_{i, j=1, 2, 3}$. – Giuseppe Negro Jun 11 '12 at 8:48
The long answer involves tensor analysis and you can read about it on books such as Itskov, Tensor algebra and tensor analysis for engineers. – Giuseppe Negro Jun 11 '12 at 8:49
Thanks, @GiuseppeNegro! – fred Jun 11 '12 at 11:49
Another possible explanation is that the dot is missing between $\nabla$ and $\vec v$, and the "gradient" is actually divergence. – user31373 Jun 11 '12 at 14:27

4 Answers 4

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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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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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. – awllower Mar 31 '13 at 8:24

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 ).

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