Stoke’s Theorem Formula
The general formula for Stoke’s Theorem in three dimensions is:
[Tex]\int\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}[/Tex]
Where:
- [Tex]\nabla \times \mathbf{F}[/Tex] represents the curl of the vector field F.
- [Tex]d\mathbf{S}[/Tex] is the vector area element of the surface S.
- [Tex]C[/Tex] is the closed curve that is the boundary of S.
- [Tex]d\mathbf{r}[/Tex] is the line element along C
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