Stoke’s Theorem Proof
According to Stoke’s Theorem, the line integral of vector field around a closed curve is equal to the surface integral of the curl of the vector field over the surface enclosed by the curve. To prove this let us denote the following
- C is the Curve whose parameter is r(t) for a ≤ t ≤ b
- S is the surface enclosed by C
- D is the region in the xy plane projected from surface S
- F(x, y, z) = (P(x, y, z), Q(x, y, z), R(x, y, z)) is the vector field defined on region S
Line integral around the closed curve C can be expressed as:
[Tex] \oint_{C}F.dr = \int_{a}^{b}F(r(t)).r'(t)dt[/Tex]
The surface integral of curl of F over S is expressed as
[Tex]\int \int_S (\bigtriangledown \times F).ndA[/Tex]
where n is the unit normal vector to the surface S, and dA is the area element on the surface S.
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