Stoke’s Theorem in Different Coordinate Systems
Stoke’s Theorem can be expressed in following different coordinate system
- Cartesian Coordinate
- Cylindrical Coordinate
- Spherical Coordinate
Stoke’s Theorem in Cartesian Coordinates
In Cartesian coordinates, the curl and the surface integral are expressed in terms of i, j, k unit vectors and the differential elements [Tex]dx, dy, dz[/Tex].
[Tex]\nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} – \frac{\partial Q}{\partial z}\right) \mathbf{i} + \left(\frac{\partial P}{\partial z} – \frac{\partial R}{\partial x}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} – \frac{\partial P}{\partial y}\right) \mathbf{k}[/Tex]
where [Tex]\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}[/Tex] is the vector field.
Stoke’s Theorem in Cylindrical Coordinates
Stoke’s Theorem in cylindrical coordinates involves the unit vectors [Tex]\mathbf{e}_\rho,\mathbf{e}_\phi, \mathbf{e}_z[/Tex] and the differentials [Tex]dρ, dφ, dz[/Tex].
The curl and the surface element are:
[Tex]\nabla \times \mathbf{F} = \left( \frac{1}{r}\frac{\partial R}{\partial \theta} – \frac{\partial Q}{\partial z} \right) \mathbf{e}r + \left( \frac{\partial P}{\partial z} – \frac{\partial R}{\partial r} \right) \mathbf{e}\theta + \frac{1}{r}\left( \frac{\partial (rQ)}{\partial r} – \frac{\partial P}{\partial \theta} \right) \mathbf{e}_z [/Tex]
[Tex]d\mathbf{S} = r dz d\theta \mathbf{e} _r + dr dz \mathbf{e}_\theta + r dr d\theta \mathbf{e}_z[/Tex]
Stoke’s Theorem in Spherical Coordinates
In spherical coordinates, the theorem uses the unit vectors [Tex]\mathbf{e}_r,\mathbf{e}_\theta,\mathbf{e}_\phi[/Tex] and the differentials [Tex]dr, dθ, dφ[/Tex].
The curl and the surface element are:
[Tex]\nabla \times \mathbf{F} = \frac{1}{\rho^2 \sin\phi}\left[ \frac{\partial}{\partial \phi} (\sin\phi R) – \frac{\partial Q}{\partial \theta} \right] \mathbf{e}\rho + \frac{1}{\rho \sin\phi}\left[ \frac{\partial P}{\partial \theta} – \frac{\partial}{\partial \rho}(\rho R) \right] \mathbf{e}\phi + \frac{1}{\rho}\left[ \frac{\partial}{\partial \rho}(\rho Q) – \frac{\partial P}{\partial \phi} \right] \mathbf{e}\theta[/Tex]
[Tex]d\mathbf{S} = \rho^2 \sin\phi d\phi d\theta \mathbf{e}\rho + \rho \sin\phi d\rho d\theta \mathbf{e}\phi + \rho d\rho d\phi \mathbf{e}\theta [/Tex]
In each coordinate system, the theorem connects the circulation of the vector field [Tex]\mathbf{F} [/Tex] along a closed curve C (the boundary of the surface S) with the flux of the curl of ( \mathbf{F} ) through the surface S. The specific form of the curl and the surface element [Tex]d\mathbf{S}[/Tex] will depend on the chosen coordinate system.
Stoke’s Theorem
Stokes’ Theorem is a fundamental result in vector calculus that relates a surface integral over a closed surface to a line integral around its boundary. It is named after the Irish mathematician Sir George Stokes, who formulated it in the 19th century. Stokes’ Theorem states that the circulation (or line integral) of a vector field around a closed curve is proportional to the flux (or surface integral) of the vector field’s curl over the surface encompassed by the curve.
In this article, we will learn in detail about Stoke’s Theorem, its formula, its expression in different coordinate system and its application.
Table of Content
- What is Stoke’s Theorem?
- Stoke’s Theorem Formula
- Stoke’s Theorem Proof
- Stoke’s Theorem vs Gauss’s Theorem
- Applications of Stoke’s Theorem