Homotopy invariance of line integrals of closed forms

Builds on Curl Line integral of a vector field Closed curve Stokes' theorem Simply connected domain Conservative vector field

Statement

Let $U \subseteq \mathbb{R}^3$ be open, and let $F \colon U \to \mathbb{R}^3$ be a vector field of class $C^1$ with $\nabla \times F = 0$ on $U$ . Let $c_0, c_1 \colon [0, 1] \to U$ be piecewise smooth closed curves. Suppose that there is a piecewise smooth map $H \colon [0, 1] \times [0, 1] \to U$ with $H(t, 0) = c_0(t)$ , $H(t, 1) = c_1(t)$ and $H(0, s) = H(1, s)$ for all $s, t \in [0, 1]$ . Then $\oint_{c_0} F \cdot d\mathbf{r} = \oint_{c_1} F \cdot d\mathbf{r}.$

Remarks

Proof: apply Stokes' theorem to the surface $H([0, 1]^2)$ . The curl term is zero, and the integrals along the curves $s \mapsto H(0, s)$ and $s \mapsto H(1, s)$ cancel. If $c_1$ is a constant loop, then $\oint_{c_0} F \cdot d\mathbf{r} = 0$ . On a simply connected domain, each loop deforms to a point, and a smoothing argument gives a piecewise smooth $H$ . So curl-free $C^1$ fields on such domains are conservative. The condition $\nabla \times F = 0$ means that the 1-form $F_1 \, dx + F_2 \, dy + F_3 \, dz$ is closed. This explains the name of the theorem.

Reading path

To understand this page from the ground up, read these 61 pages in order. The foundations come first.

Euclidean space object
Function concept
Function composition concept
Infimum concept
Supremum concept
Fréchet derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Jordan measure definition
Limit of a function of several variables concept
Differentiable function of several variables concept
Parametrised curve concept
Iterated integral definition
Partial derivative concept
Continuity in R^n definition
Differentiable function of several variables definition
Partial derivative definition
Gradient concept
Chain rule in several variables statement
Differentiable on an open set definition
Double integral definition
Gradient definition
Higher partial derivatives definition
Jacobian matrix definition
Chain rule for a composition with a curve statement
Fubini's theorem statement
Iterated derivative definition
Triple integral definition
C^k function characterised by continuous derivatives statement
Differential form definition
Parametrised curve definition
Vector field definition
Wedge product of forms definition
Closed curve definition
Parametrised surface definition
Rectifiable curve definition
Conservative vector field definition
Curl definition
Tangent vector of a curve definition
Divergence definition
Simply connected domain definition
Line integral of a scalar field definition
Line integral of a vector field definition
Tangent plane and normal vector of a surface definition
Regular curve definition
Green's theorem statement
One-form definition
Surface area definition
Arc length parametrisation definition
Surface integral definition
Divergence theorem on a rectangle in the plane statement
Exterior derivative definition
Line integral as an integral over the parameter interval statement
Orientable surface definition
Gradient theorem statement
Flux through a surface definition
Divergence theorem statement
Generalised Stokes theorem statement
Stokes' theorem statement
Homotopy invariance of line integrals of closed forms statement

Tag	`0YVVN6R715`
URI	`m2://0YVVN6R715`
Aliases	`helmholtz-theorem-for-lamellar-fields`
Kind	statement, theorem
Writing	W0 stub
MSC	26B20
Status	This copy of the site has no build data. Run `m2 status 0YVVN6R715`.
Uses	Curl, Line integral of a vector field, Closed curve, Stokes' theorem, Simply connected domain, Conservative vector field

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