### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# Contents

## Definition

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

The gradient of $f$ is the vector field

$\nabla f := g^{-1} d_{dR} f \in \Gamma(T X) \,,$

where $d_{dR} : C^\infty(X) \to \Omega^1(X)$ is the de Rham differential.

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

$d_{dR} f = g(-,\nabla f)$

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

$d_{dR} f = \star_g \iota_{\nabla f} vol_g \,,$

where $vol_g$ 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 $vol$ 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

$grad A = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} A d S$

where $D$ runs over the domains with smooth boundary $\partial D$ containing point $x$ and $\vec{n}$ 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 $(M,g)$ is the Cartesian space $\mathbb{R}^n$ endowed with the standard Euclidean metric, then

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

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)