24.1 Statement and proof of Stokes' theorem

Stokes' theorem connects surface integrals of curl to line integrals around boundaries. It's a powerful tool in vector calculus, generalizing earlier theorems and finding applications in physics and engineering.

Understanding Stokes' theorem requires familiarity with vector fields, differential forms, and various types of integrals. It's a key concept that ties together many ideas in multivariable calculus and vector analysis.

Vector Calculus Fundamentals

Vector Fields and Differential Forms

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• Vector field $\mathbf{F}(x, y, z) = (F_1(x, y, z), F_2(x, y, z), F_3(x, y, z))$ assigns a vector to each point in a subset of space
• Curl of a vector field $\mathbf{F}$, denoted $\nabla \times \mathbf{F}$, measures the infinitesimal rotation of the field
  • In 3D, $\nabla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}, \frac{\partial F_1}{\partial z} - \frac{\partial F_3}{\partial x}, \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y}\right)$
  • Useful for understanding the rotational behavior of a vector field (fluid dynamics, electromagnetism)
• Differential form $\omega$ is an object that can be integrated over oriented manifolds (curves, surfaces, volumes)
  • 1-form $\omega = P\,dx + Q\,dy + R\,dz$ can be integrated over curves
  • 2-form $\omega = P\,dy \wedge dz + Q\,dz \wedge dx + R\,dx \wedge dy$ can be integrated over surfaces
• Exterior derivative $d\omega$ generalizes the gradient, curl, and divergence operations to differential forms
  • For a 1-form $\omega = P\,dx + Q\,dy + R\,dz$, $d\omega = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)dy \wedge dz + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)dz \wedge dx + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)dx \wedge dy$

Integration Concepts

Surface and Line Integrals

• Surface integral $\iint_S \mathbf{F} \cdot d\mathbf{S}$ evaluates a vector field $\mathbf{F}$ over a surface $S$
  • $d\mathbf{S} = \mathbf{n}\,dS$, where $\mathbf{n}$ is the unit normal vector to the surface and $dS$ is the surface area element
  • Measures the flux of a vector field through a surface (fluid flow, electric flux)
• Line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ evaluates a vector field $\mathbf{F}$ along a curve $C$
  • $d\mathbf{r} = (dx, dy, dz)$ is the infinitesimal displacement vector along the curve
  • Measures the work done by a force field along a path (work, circulation)

Orientable Surfaces and Boundary Curves

• Orientable surface is a surface that has a consistent notion of "clockwise" and "counterclockwise"
  • Examples: sphere, torus, plane
  • Non-orientable surfaces (Möbius strip, Klein bottle) do not have a consistent orientation
• Boundary curve $\partial S$ is the oriented curve that forms the edge of an oriented surface $S$
  • Orientation of $\partial S$ is determined by the right-hand rule with respect to the surface normal
  • Important for relating line integrals and surface integrals via Stokes' theorem

Stokes' Theorem

Statement and Applications of Stokes' Theorem

• Stokes' theorem relates the surface integral of the curl of a vector field over a surface $S$ to the line integral of the vector field over the boundary curve $\partial S$
  • $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$
  • Generalizes the Fundamental Theorem of Calculus, Green's Theorem, and the Divergence Theorem
• Stokes' theorem has numerous applications in physics and engineering
  • Faraday's law of induction: EMF induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop
  • Kelvin's circulation theorem: circulation of a velocity field around a closed curve is equal to the surface integral of vorticity over any surface bounded by the curve
• To apply Stokes' theorem:
  1. Identify the vector field $\mathbf{F}$, surface $S$, and boundary curve $\partial S$
  2. Compute the curl $\nabla \times \mathbf{F}$
  3. Parametrize the surface $S$ and compute the surface integral $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}$
  4. Parametrize the boundary curve $\partial S$ and compute the line integral $\oint_{\partial S} \mathbf{F} \cdot d\mathbf{r}$
  5. Verify that the surface and line integrals are equal