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.
What does it mean to take the gradient of a vector field?
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multivariable-calculus
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2The 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 $(\partial_i v_j)_{i, j=1, 2, 3}$. – 2012-06-11
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1The long answer involves tensor analysis and you can read about it on books such as Itskov, *Tensor algebra and tensor analysis for engineers*. – 2012-06-11
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1Another possible explanation is that the dot is missing between $\nabla$ and $\vec v$, and the "gradient" is actually divergence. – 2012-06-11
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1Guiseppe Negro's short answer is off, switch his i's and j's and its fixed. – 2013-05-21
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0@Thomas I've converted your answer into a comment. In the future please only use answers to give answers to the question posed in the original post. I realize that you do not yet have the reputation to comment on other people's posts, but that will come before long if you contribute to the site. – 2013-05-21
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0and in the special case of the vector being the electromagnetic vector field $\vec A$, is $\vec{\nabla} \vec{A} =0$? – 2018-03-21