Question

State Stokes's Theorem.

Short answer

Stokes' Theorem states that the integral of the curl of a vector field over a surface is equal to the line integral of the vector field along the boundary of the surface. Mathematical expression: \( \int_{S} (\nabla \times \vec{F})\.d\vec{S} = \oint_{\partial S} \vec{F} \cdot d\vec{r} \)

Step-by-step solution

Introduction of Stokes' Theorem

Stokes' Theorem is a statement about the integration of differential forms. In simple terms, it's a method to transform a surface integral of a vector field into a line integral on the boundary of the surface.

Statement of Stokes' Theorem

The formal statement of Stokes' Theorem in its most basic form is: The surface integral of the curl of a vector field over a surface S is equal to the line integral of the vector field over the boundary of S. Mathematically, this is written as: \[ \int_{S} (\nabla \times \vec{F})\.d\vec{S} = \oint_{\partial S} \vec{F} \cdot d\vec{r} \] Where: \( \vec{F} \) is a vector field, \( S \) is a surface, \( \partial S \) is the boundary of \( S \), and \( \nabla \times \vec{F} \) is the curl of \( \vec{F} \).

Explanation of the terms in the formula

In this formula, on the left side we have a surface integral of the curl of a vector field, which could represent the rotation of a vector field. The right side consists of a line integral that captures the magnitude and direction of the field along the boundary of the surface.