What Is Curl in Higher-Dimensional Spaces?

hfwang132@gmail.com (hfwang132) — Fri, 26 May 2023 21:00:37 +0800

Differential Forms

Before introducing curl, we must first introduce differential forms and the exterior derivative operator.

An n-form can be defined as an alternating multilinear map \(\omega:(T_pM)^n\rightarrow \mathbb{R}\) . It maps multiple vectors to a real number. Moreover, it satisfies alternation: exchanging two input vectors introduces an additional minus sign in the output.

Thus, an n-form can be explicitly defined as follows:

\[\omega^1\wedge \omega^2\wedge\cdots\wedge \omega^n(v_1,v_2,\cdots,v_n)= \begin{vmatrix} \omega^1(v_1) & \cdots & \omega^{1}(v_n) \\ \vdots & \ddots & \vdots \\ \omega^n(v_1) & \cdots & \omega^n(v_n) \end{vmatrix}\in\mathbb{R}\]

The set of all n-forms on \(T_pM\) can be written as \(\bigwedge^n(T^*_pM)\) .

A Concise Proof of Vector Product Rules

hfwang132@gmail.com (hfwang132) — Sat, 27 Feb 2021 00:46:19 +0800

I believe that anyone learning vector calculus for the first time finds the product rules for vectors quite daunting:

Easy difficulty:

\[\nabla(fg)=f\nabla g+g\nabla f\]\[\nabla\cdot(f\mathrm{A})=f\nabla\cdot\mathrm{A}+\nabla f\cdot \mathrm{A}\]\[\nabla\times(f\mathrm{A})=f\nabla\times\mathrm{A}+\nabla f\times\mathrm{A}\]

Hard difficulty:

\[\nabla\cdot(\mathrm{A}\times\mathrm{B})=\mathrm{B}\cdot(\nabla\times\mathrm{A})-\mathrm{A}\cdot(\nabla\times\mathrm{B})\]

Hell difficulty:

\[\nabla(\mathrm{A}\cdot \mathrm{B})=\mathrm{A}\times(\nabla\times\mathrm{B})+(\mathrm{A}\cdot\nabla)\mathrm{B}+\mathrm{B}\times(\nabla\times\mathrm{A})+(\mathrm{B}\cdot\nabla)\mathrm{A}\]\[\nabla\times(\mathrm{A}\times\mathrm{B})=\mathrm{A}(\nabla\cdot \mathrm{B})-\mathrm{B}(\nabla\cdot\mathrm{A})+(\mathrm{B}\cdot\nabla)\mathrm{A}-(\mathrm{A}\cdot\nabla)\mathrm{B}\]

DLC: second derivatives

\[\nabla\times(\nabla f)=0\]\[\nabla\cdot(\nabla\times\mathrm{A})=0\]\[\nabla\times(\nabla\times\mathrm{A})=\nabla(\nabla\cdot\mathrm{A})-(\nabla\cdot\nabla)\mathrm{A}\]

A Concise Proof

To make the proof as concise as possible, we would like to use a single expression with subscripts to represent multiple expressions, rather than having to write out x, y, and z every time.

Vector Calculus on Haifei's Home

What Is Curl in Higher-Dimensional Spaces?

Differential Forms

A Concise Proof of Vector Product Rules

A Concise Proof