Geometric calculus identities

See also: Geometric calculus

The following are important and widely-used identities involving derivatives and integrals in geometric calculus.

First derivative identities

Distributive rules

For a vector $\overrightarrow{x}$ in a geometric algebra of (non-degenerate) dimension $n$, $\mathrm{\nabla }\overrightarrow{x}=n$, and $\mathrm{\nabla }\wedge \overrightarrow{x}=0$. If $\overrightarrow{a}$ is a constant vector with respect to $\overrightarrow{x}$, then

$\begin{array}{rl}\mathrm{\nabla }\overrightarrow{a}& =0,\\ \mathrm{\nabla }\left(\overrightarrow{x}\overrightarrow{a}\right)& =n\overrightarrow{a},\\ \mathrm{\nabla }(\overrightarrow{x}\cdot \overrightarrow{a})& =(\overrightarrow{a}\cdot \mathrm{\nabla })\overrightarrow{x}=\overrightarrow{a},\\ \mathrm{\nabla }(\overrightarrow{x}\wedge \overrightarrow{a})& =\mathrm{\nabla }\cdot (\overrightarrow{x}\wedge \overrightarrow{a})=n\overrightarrow{a}-\overrightarrow{a}.\end{array}$

The first two rules above also hold if $\overrightarrow{a}$ is replaced with any multivector $A$ that does not depend on $\overrightarrow{x}$.

The vector derivative, like the standard derivative, is also distributive over addition. That is, for any arbitrary multivector-valued functions $F$ and $G$,

$\mathrm{\nabla }(F+G)=\mathrm{\nabla }F+\mathrm{\nabla }G.$

Similarly, combining this with the above multiplication rule, this also holds for subtraction, where

$\mathrm{\nabla }(F-G)=\mathrm{\nabla }F-\mathrm{\nabla }G.$

Product rule

The product rule in geometric algebra can be stated as follows:

$\mathrm{\nabla }\left(FG\right)=(\mathrm{\nabla }F)G+\dot{\mathrm{\nabla }}F\dot{G}$

or, when $F$ commutes with all vectors in the algebra,

$\mathrm{\nabla }\left(FG\right)=(\mathrm{\nabla }F)G+F(\mathrm{\nabla }G),$

analogously to the product rule in single-variable calculus.

Chain rule

For a scalar-valued function $\lambda \left(\overrightarrow{x}\right)$ and an arbitrary multivector-valued function $F$,

${\mathrm{\nabla }}_{\overrightarrow{x}}F\left(\lambda \left(\overrightarrow{x}\right)\right)=\left({\mathrm{\nabla }}_{\overrightarrow{x}}\lambda \right)\frac{\partial F}{\partial \lambda },$

and for a vector-valued function $\overrightarrow{y}\left(\overrightarrow{x}\right)$ and an arbitrary multivector-valued function $F$,

${\mathrm{\nabla }}_{\overrightarrow{x}}F\left(\overrightarrow{y}\left(\overrightarrow{x}\right)\right)=(\left({\mathrm{\nabla }}_{\overrightarrow{x}}\overrightarrow{y}\right)\cdot {\mathrm{\nabla }}_{\overrightarrow{y}})F\left(\overrightarrow{y}\right),$

analogous to the chain rule commonly seen with pure scalar functions.

List of derivatives for common functions

Vector derivatives of functions with vector-valued inputs

Table of common functions

function $F(\overrightarrow{x})$	$\mathrm{\nabla }F(\overrightarrow{x})$	$\mathrm{\nabla }\cdot F(\overrightarrow{x})$	$\mathrm{\nabla }\wedge F(\overrightarrow{x})$
$\overrightarrow{x}$	$n$	$n$	$0$
$\overrightarrow{x}\cdot \overrightarrow{a}$	$\overrightarrow{a}$	$0$	$\overrightarrow{a}$
$\overrightarrow{x}A$	$nA$
$|\overrightarrow{x}|$	$\hat{x}$	$\hat{x}$	$0$
$|\overrightarrow{x}{|}^k$	$k|\overrightarrow{x}{|}^{k-1}\hat{x}$	$k|\overrightarrow{x}{|}^{k-1}\hat{x}$	$0$
$|\overrightarrow{x}{|}^k\overrightarrow{x}$	$(2+k)|\overrightarrow{x}{|}^k$	$(2+k)|\overrightarrow{x}{|}^k$	$0$
$G(|\overrightarrow{x}|)$	$\hat{x}\frac{\partial G}{\partial |\overrightarrow{x}|}$
$G(\overrightarrow{x}-{\overrightarrow{x}}^{\prime })$	$\mathrm{\nabla }G$	$\mathrm{\nabla }\cdot G$	$\mathrm{\nabla }\wedge G$

List of common identities

Gradient

• $\mathrm{\nabla }(\psi +\varphi )=\mathrm{\nabla }\psi +\mathrm{\nabla }\varphi$
• $\mathrm{\nabla }(\psi \varphi )=\psi (\mathrm{\nabla }\varphi )+\varphi (\mathrm{\nabla }\psi )$
• $\mathrm{\nabla }\wedge \varphi =\mathrm{\nabla }\varphi ,\mathrm{\nabla }\cdot \varphi =0$
• $\mathrm{\nabla }(\overrightarrow{u}\cdot \overrightarrow{v})=(\overrightarrow{u}\cdot \mathrm{\nabla })\overrightarrow{v}+(\overrightarrow{v}\cdot \mathrm{\nabla })\overrightarrow{u}-\overrightarrow{u}\cdot (\mathrm{\nabla }\wedge \overrightarrow{v})-\overrightarrow{v}\cdot (\mathrm{\nabla }\wedge \overrightarrow{u})$

Interior derivative

• $\mathrm{\nabla }\cdot (\varphi \overrightarrow{v})=(\mathrm{\nabla }\varphi )\cdot \overrightarrow{v}+\varphi (\mathrm{\nabla }\cdot \overrightarrow{v})$
• $\mathrm{\nabla }\cdot (\overrightarrow{u}\wedge \overrightarrow{v})=(\mathrm{\nabla }\cdot \overrightarrow{u})\overrightarrow{v}+(\overrightarrow{u}\cdot \mathrm{\nabla })\overrightarrow{v}-(\mathrm{\nabla }\cdot \overrightarrow{v})\overrightarrow{u}-(\overrightarrow{v}\cdot \mathrm{\nabla })\overrightarrow{u}$

Exterior derivative

• $\mathrm{\nabla }\wedge (\varphi \overrightarrow{v})=(\mathrm{\nabla }\varphi )\wedge \overrightarrow{v}+\varphi (\mathrm{\nabla }\wedge \overrightarrow{v})$
• $\mathrm{\nabla }\wedge (\overrightarrow{u}\cdot \overrightarrow{v})=\mathrm{\nabla }(\overrightarrow{u}\cdot \overrightarrow{v})$
• $\mathrm{\nabla }\wedge (\overrightarrow{u}\wedge \overrightarrow{v})=\overrightarrow{v}\wedge (\mathrm{\nabla }\wedge \overrightarrow{u})-\overrightarrow{u}\wedge (\mathrm{\nabla }\wedge \overrightarrow{v})$

Second derivatives

• $\mathrm{\nabla }\wedge (\mathrm{\nabla }\varphi )=0$
• $\mathrm{\nabla }\wedge (\mathrm{\nabla }\wedge F)=0$
• $\mathrm{\nabla }\cdot (\mathrm{\nabla }\cdot F)=0$
• $\mathrm{\nabla }\cdot (\mathrm{\nabla }\varphi )={\mathrm{\nabla }}^2\varphi$
• ${\mathrm{\nabla }}^{2}F=\mathrm{\nabla }(\mathrm{\nabla }\cdot F)+\mathrm{\nabla }(\mathrm{\nabla }\wedge F)$ (multivector Laplacian)

Multivector derivatives of functions with multivector-valued inputs

Table of common functions

function $F(X)$	${\mathrm{\nabla }}_{X}F(X)$
$X$	$n$
$|X{|}^k$	$k|X{|}^{k-2}\tilde{X}$
$|X{|}^{k}X$	$k|X{|}^{k-2}\tilde{X}X+n|X{|}^k$
$\ln (|X|)$	$\frac{\tilde{X}}{|X{|}^2}$
$G(|X|)$	$\frac{\tilde{X}}{|X|}\frac{\partial G}{\partial |X|}$

References

This article "Geometric calculus identities" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Geometric calculus identities. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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