Exterior derivative
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On a
If a differential k-form is thought of as measuring the flux through an infinitesimal k-parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope at each point.
Definition
The exterior derivative of a differential form of degree k (also differential k-form, or just k-form for brevity here) is a differential form of degree k + 1.
If f is a
The exterior product of differential forms (denoted with the same symbol ∧) is defined as their
There are a variety of equivalent definitions of the exterior derivative of a general k-form.
In terms of axioms
The exterior derivative is defined to be the unique ℝ-linear mapping from k-forms to (k + 1)-forms that has the following properties:
- df is the differential of f for a 0-form f .
- d(df ) = 0 for a 0-form f .
- d(α ∧ β) = dα ∧ β + (−1)p (α ∧ dβ) where α is a p-form. That is to say, d is an antiderivation of degree 1 on the exterior algebra of differential forms (see the graded product rule).
The second defining property holds in more generality: d(dα) = 0 for any k-form α; more succinctly, d2 = 0. The third defining property implies as a special case that if f is a function and α is a k-form, then d( fα) = d( f ∧ α) = df ∧ α + f ∧ dα because a function is a 0-form, and scalar multiplication and the exterior product are equivalent when one of the arguments is a scalar.[citation needed]
In terms of local coordinates
Alternatively, one can work entirely in a
over ℝn is defined as
(using the
where each of the components of the multi-index I run over all the values in {1, ..., n}. Note that whenever i equals one of the components of the multi-index I then dxi ∧ dxI = 0 (see
The definition of the exterior derivative in local coordinates follows from the preceding definition in terms of axioms. Indeed, with the k-form φ as defined above,
Here, we have interpreted g as a 0-form, and then applied the properties of the exterior derivative.
This result extends directly to the general k-form ω as
In particular, for a 1-form ω, the components of dω in
Caution: There are two conventions regarding the meaning of . Most current authors[citation needed] have the convention that
while in older text like Kobayashi and Nomizu or Helgason
In terms of invariant formula
Alternatively, an explicit formula can be given [1] for the exterior derivative of a k-form ω, when paired with k + 1 arbitrary smooth vector fields V0, V1, ..., Vk:
where [Vi, Vj] denotes the Lie bracket and a hat denotes the omission of that element:
In particular, when ω is a 1-form we have that dω(X, Y) = dX(ω(Y)) − dY(ω(X)) − ω([X, Y]).
Note: With the conventions of e.g., Kobayashi–Nomizu and Helgason the formula differs by a factor of 1/k + 1:
Examples
Example 1. Consider σ = u dx1 ∧ dx2 over a 1-form basis dx1, ..., dxn for a scalar field u. The exterior derivative is:
The last formula, where summation starts at i = 3, follows easily from the properties of the
Example 2. Let σ = u dx + v dy be a 1-form defined over ℝ2. By applying the above formula to each term (consider x1 = x and x2 = y) we have the sum
Stokes' theorem on manifolds
If M is a compact smooth orientable n-dimensional manifold with boundary, and ω is an (n − 1)-form on M, then
Intuitively, if one thinks of M as being divided into infinitesimal regions, and one adds the flux through the boundaries of all the regions, the interior boundaries all cancel out, leaving the total flux through the boundary of M.
Further properties
Closed and exact forms
A k-form ω is called closed if dω = 0; closed forms are the kernel of d. ω is called exact if ω = dα for some (k − 1)-form α; exact forms are the image of d. Because d2 = 0, every exact form is closed. The Poincaré lemma states that in a contractible region, the converse is true.
de Rham cohomology
Because the exterior derivative d has the property that d2 = 0, it can be used as the
Naturality
The exterior derivative is natural in the technical sense: if f : M → N is a smooth map and Ωk is the contravariant smooth functor that assigns to each manifold the space of k-forms on the manifold, then the following diagram commutes
so d( f∗ω) = f∗dω, where f∗ denotes the pullback of f . This follows from that f∗ω(·), by definition, is ω( f∗(·)), f∗ being the pushforward of f . Thus d is a natural transformation from Ωk to Ωk+1.
Exterior derivative in vector calculus
Most vector calculus operators are special cases of, or have close relationships to, the notion of exterior differentiation.
Gradient
A
When an inner product ⟨·,·⟩ is defined, the gradient ∇f of a function f is defined as the unique vector in V such that its inner product with any element of V is the directional derivative of f along the vector, that is such that
That is,
where ♯ denotes the musical isomorphism ♯ : V∗ → V mentioned earlier that is induced by the inner product.
The 1-form df is a section of the cotangent bundle, that gives a local linear approximation to f in the cotangent space at each point.
Divergence
A vector field V = (v1, v2, ..., vn) on ℝn has a corresponding (n − 1)-form
where denotes the omission of that element.
(For instance, when n = 3, i.e. in three-dimensional space, the 2-form ωV is locally the
of V over that hypersurface.The exterior derivative of this (n − 1)-form is the n-form
Curl
A vector field V on ℝn also has a corresponding 1-form
Locally, ηV is the
When n = 3, in three-dimensional space, the exterior derivative of the 1-form ηV is the 2-form
Invariant formulations of operators in vector calculus
The standard vector calculus operators can be generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows:
where ⋆ is the
Note that the expression for curl requires ♯ to act on ⋆d(F♭), which is a form of degree n − 2. A natural generalization of ♯ to k-forms of arbitrary degree allows this expression to make sense for any n.
See also
Notes
- ^ Spivak(1970), p 7-18, Th. 13
References
- JFM 30.0313.04. Retrieved 2 Feb 2016.
- Conlon, Lawrence (2001). Differentiable manifolds. Basel, Switzerland: Birkhäuser. p. 239. ISBN 0-8176-4134-3.
- Darling, R. W. R. (1994). Differential forms and connections. Cambridge, UK: Cambridge University Press. p. 35. ISBN 0-521-46800-0.
- Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. p. 20. ISBN 0-486-66169-5.
- Loomis, Lynn H.; Sternberg, Shlomo (1989). Advanced Calculus. Boston: Jones and Bartlett. pp. 304–473 (ch. 7–11). ISBN 0-486-66169-5.
- Ramanan, S. (2005). Global calculus. Providence, Rhode Island: American Mathematical Society. p. 54. ISBN 0-8218-3702-8.
- ISBN 9780805390216.
- ISBN 0-914098-00-4
- Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer, ISBN 0-387-90894-3
External links
- Archived at Ghostarchive and the Wayback Machine: "The derivative isn't what you think it is". Aleph Zero. November 3, 2020 – via YouTube.