### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

Let $\left(X,g\right)$ be a Riemannian manifold and $f\in {C}^{\infty }\left(X\right)$ a function.

The gradient of $f$ is the vector field

$\nabla f:={g}^{-1}{d}_{\mathrm{dR}}f\in \Gamma \left(TX\right)\phantom{\rule{thinmathspace}{0ex}},$\nabla f := g^{-1} d_{dR} f \in \Gamma(T X) \,,

where ${d}_{\mathrm{dR}}:{C}^{\infty }\left(X\right)\to {\Omega }^{1}\left(X\right)$ is the de Rham differential.

This is the unique vector field $\nabla f$ such that

${d}_{\mathrm{dR}}f=g\left(-,\nabla f\right)$d_{dR} f = g(-,\nabla f)

or equivalently, if the manifold is oriented, this is the unique vector field such that

${d}_{\mathrm{dR}}f={\star }_{g}{\iota }_{\nabla f}{\mathrm{vol}}_{g}\phantom{\rule{thinmathspace}{0ex}},$d_{dR} f = \star_g \iota_{\nabla f} vol_g \,,

where ${\mathrm{vol}}_{g}$ is the volume form and ${\star }_{g}$ is the Hodge star operator induced by $g$. (The result is independent of orientation, which can be made explicit by interpreting both $\mathrm{vol}$ and $\star$ as valued in pseudoforms.)

Alternatively, the gradient of a scalar field $A$ in some point $x\in M$ is calculated (or alternatively defined) by the integral formula

$\mathrm{grad}A={\mathrm{lim}}_{\mathrm{vol}D\to 0}\frac{1}{\mathrm{vol}D}{\oint }_{\partial D}\stackrel{⇀}{n}AdS$grad A = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} A d S

where $D$ runs over the domains with smooth boundary $\partial D$ containing point $x$ and $\stackrel{⇀}{n}$ is the unit vector of outer normal to the surface $S$. The formula does not depend on the shape of boundaries taken in limiting process, so one can typically take a coordinate chart and balls with decreasing radius in this particular coordinate chart.

## Example

If $\left(M,g\right)$ is the Cartesian space ${ℝ}^{n}$ endowed with the standard Euclidean metric, then

$\nabla f=\sum _{i=1}^{n}\frac{\partial f}{\partial {x}^{i}}{\partial }_{i}.$\nabla f= \sum_{i=1}^n\frac{\partial f}{\partial x^i}\partial_i .

This is the classical gradient from vector analysis?.

## Remark

In many classical applications of the gradient in vector analysis?, the Riemannian structure is actually irrelevant, and the gradient can be replaced with the differential 1-form.

Revised on September 5, 2011 18:11:45 by Toby Bartels (75.88.82.16)