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gradient

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Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

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Definition

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

The gradient of ff is the vector field

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

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

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

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

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

d dRf= gι fvol g, d_{dR} f = \star_g \iota_{\nabla f} vol_g \,,

where vol gvol_g is the volume form and g\star_g is the Hodge star operator induced by gg. (The result is independent of orientation, which can be made explicit by interpreting both volvol and \star as valued in pseudoforms.)

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

gradA=lim volD01volD DnAdS grad A = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} A d S

where DD runs over the domains with smooth boundary D\partial D containing point xx and n\vec{n} is the unit vector of outer normal to the surface SS. 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)(M,g) is the Cartesian space n\mathbb{R}^n endowed with the standard Euclidean metric, then

f= i=1 nfx i 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)