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Exterior Derivative


The exterior derivative of a function f is the one-form

 df=sum_(i)(partialf)/(partialx_i)dx_i
(1)

written in a coordinate chart (x_1,...,x_n). Thinking of a function as a zero-form, the exterior derivative extends linearly to all differential k-forms using the formula

 d(alpha ^ beta)=dalpha ^ beta+(-1)^kalpha ^ dbeta,
(2)

when alpha is a k-form and where  ^ is the wedge product.

The exterior derivative of a k-form is a (k+1)-form. For example, for a differential k-form

 omega^1=b_1dx_1+b_2dx_2,
(3)

the exterior derivative is

 domega^1=db_1 ^ dx_1+db_2 ^ dx_2.
(4)

Similarly, consider

 omega^1=b_1(x_1,x_2)dx_1+b_2(x_1,x_2)dx_2.
(5)

Then

domega^1=db_1 ^ dx_1+db_2 ^ dx_2
(6)
=((partialb_1)/(partialx_1)dx_1+(partialb_1)/(partialx_2)dx_2) ^ dx_1+((partialb_2)/(partialx_1)dx_1+(partialb_2)/(partialx_2)dx_2) ^ dx_2.
(7)

Denote the exterior derivative by

 Dt=partial/(partialx) ^ t.
(8)

Then for a 0-form t,

 (Dt)_mu=(partialt)/(partialx^mu),
(9)

for a 1-form t,

 (Dt)_(munu)=1/2((partialt_nu)/(partialx^mu)-(partialt_mu)/(partialx^nu)),
(10)

and for a 2-form t,

 (Dt)_(ijk)=1/3epsilon_(ijk)((partialt_(23))/(partialx^1)+(partialt_(31))/(partialx^2)+(partialt_(12))/(partialx^3)),
(11)

where epsilon_(ijk) is the permutation tensor.

It is always the case that d(dalpha)=0. When dalpha=0, then alpha is called a closed form. A top-dimensional form is always a closed form. When alpha=deta then alpha is called an exact form, so any exact form is also closed. An example of a closed form which is not exact is dtheta on the circle. Since theta is a function defined up to a constant multiple of 2pi, dtheta is a well-defined one-form, but there is no function for which it is the exterior derivative.

The exterior derivative is linear and commutes with the pullback omega^* of differential k-forms omega. That is,

 df^*(alpha)=f^*(dalpha).
(12)

Hence the pullback of a closed form is closed and the pullback of an exact form is exact. Moreover, a de Rham Cohomology class [alpha] has a well-defined pullback map [f^*(alpha)].


See also

Differential k-Form, Exterior Algebra, Hodge Star, Jacobian, Manifold, Poincaré's Lemma, Stokes' Theorem, Tangent Bundle, Tensor, Wedge Product

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Exterior Derivative." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ExteriorDerivative.html

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