Stoke’s Theorem vs Gauss’s Theorem
While both theorems relate surface integrals to volume integrals, Stoke’s Theorem applies to surfaces (2D manifolds with boundary), where as Gauss Theorem applies to volumes (3D manifolds).
Aspect | Stokes’ Theorem | Gauss Divergence Theorem |
---|---|---|
Equation | [Tex]\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_{S} (\nabla \times \mathbf{F}) \cdot d\mathbf{S}[/Tex] | [Tex]\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV[/Tex] |
Integral | Relate Line integral & Surface Integral | Relate Surface integral & Volume Integral |
Boundary Integral | Closed Curve | Closed Surface |
Mathematical Focus | Curl of a Vector Field | Divergence of a Vector Field |
Physical Interpretation | Circulation along a Curve | Net Flux through a Surface |
Dimensionality | 2-Dimensional within 3-D space | 3-Dimensional |
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