Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
Did you have a look at this? –  draks ... Jun 11 '12 at 8:42
1  
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

2 Answers 2

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).

share|improve this answer
    
Hope this edit is acceptable. –  awllower Mar 31 '13 at 8:24

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.