multivariable calculus - Proof for the curl of a curl of a vector field ...
2015年1月18日 · For curl, you get a sign depending on the sign of the permutation, but you need to compute the curl twice, so you are done. Notice that it is enough to show the cases $1\leftrightarrow 2$ and $1\leftrightarrow 3$ .
Intuition on the curl formula - Mathematics Stack Exchange
Thus the component of the curl ${\rm curl}\; {\bf F} \cdot {\bf n}$ is the sum of two rates of rotation independent of the choice of ${\bf i}, {\bf j}$. One way to think of it is that the curl is twice the average of two rates of rotation. Why twice? Why the average? Plausible arguments: (1) It …
Inverse of the curl - Mathematics Stack Exchange
$\begingroup$ QED is working his way through a 1984 article in Comm. Math. Phys. by Kato, Majda, and Beale, link under my answer. Yesterday's question was about equation (7), today equation (9) $\endgroup$
Anti-curl operator - Mathematics Stack Exchange
This is called Poincare's Lemma. I will write the standard version, in a neighborhood around the origin. The usual phrasing is to say that a closed form is locally exact, the fact that this is not globally true is the stuff of cohomology.
How to derive or logically explain the formula for curl?
2017年8月12日 · Most books state that the formula for curl of a vector field is given by $\nabla \times \vec{V}$ where $\vec{V}$ is a differentiable vector field. Also, they state that: "The curl of a vector field measures the tendency for the vector field to swirl around". But, none of them state the derivation of the formula.
Visualizing the curl of curl - Mathematics Stack Exchange
2023年6月29日 · In some differential equations from phyiscs, e.g. in elastodynamics, terms with the curl of the curl of a vector field appear.
Curl of Cross Product of Two Vectors - Mathematics Stack Exchange
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Curl in cylindrical coordinates - Mathematics Stack Exchange
A correct definition of the "gradient operator" in cylindrical coordinates is \begin{equation} \nabla = e_r \frac{\partial}{\partial r} + e_\theta \frac{1}{r} \frac ...
calculus - Curl matrix operation - Mathematics Stack Exchange
$\begingroup$ That is a good insight about Df-(Df)^T having the correct components, but it's not clear to me how one would go from that matrix to the curl vector $\endgroup$ – panofsteel Commented May 13, 2023 at 1:15
Why is the divergence of curl expected to be zero?
2020年11月5日 · Then why is the divergence of the curl always zero? $$\nabla\cdot\nabla\times \bf A = 0$$ and not: $$\nabla\cdot\nabla\times \bf A =\delta^3(\bf c)$$ where $\bf c $ is a vector with speed of light magnitude. Edit: This question is about asymmetry of electromagnetic potential fields and I have no idea why it was migrated to math forum.