Explanation of Gauss's Theorem and Stokes' Theorem: Key Concepts and Examples

Two fundamental results in vector calculus, Gauss's Theorem and Stokes' Theorem, play crucial roles in understanding the relationships between surface, volume, and line integrals. These theorems are essential in various fields, particularly in physics and engineering. This article delves into the details of each theorem and provides examples to illustrate their practical applications.

What is Gauss's Theorem?

Gauss's Theorem, also known as the Divergence Theorem, relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field within that surface.

Statement

Mathematically, Gauss's Theorem is expressed as:

[ iint_{S} mathbf{F} cdot mathbf{n} , dS iiint_{V} abla cdot mathbf{F} , dV ]

where:

( S ) is a closed surface ( V ) is the volume enclosed by the surface ( S ) ( mathbf{F} ) is a vector field ( mathbf{n} ) is the outward unit normal vector on surface ( S ) ( dS ) is the differential area element on the surface ( dV ) is the differential volume element

Example

Consider the vector field ( mathbf{F} xyz ) and a cube with coordinates ( 0 leq x, y, z leq 1 ).

To calculate the divergence:

[ abla cdot mathbf{F} frac{partial xyz}{partial x} frac{partial xyz}{partial y} frac{partial xyz}{partial z} 1 cdot 1 cdot 1 3 ]

The volume integral is:

[ iiint_{V} abla cdot mathbf{F} , dV iiint_{[0,1]^3} 3 , dV 3 times 1^3 3 ]

For the surface integral, the flux through the surface can be calculated directly. However, by Gauss's Theorem, the total outward flux through the surface of the cube is also 3.

What is Stokes' Theorem?

Stokes' Theorem relates the surface integral of a vector field's curl over a surface to the line integral of the vector field around the boundary of that surface.

Statement

Mathematically, Stokes' Theorem is expressed as:

[ iint_{S} ( abla times mathbf{F}) cdot mathbf{n} , dS oint_{partial S} mathbf{F} cdot dmathbf{r} ]

where:

( S ) is an oriented surface ( partial S ) is the boundary of surface ( S ) ( mathbf{n} ) is the unit normal vector to the surface ( S ) ( dmathbf{r} ) is the differential line element along the boundary

Example

Consider the vector field ( mathbf{F} -y x , 0 ) and a surface ( S ) in the ( xy )-plane bounded by the circle ( x^2 y^2 1 ).

To calculate the curl:

[ abla times mathbf{F} begin{vmatrix} mathbf{i} mathbf{j} mathbf{k} frac{partial}{partial x} frac{partial}{partial y} frac{partial}{partial z} -y x 0 end{vmatrix} 0 - 0 mathbf{i} - 0 - 0 mathbf{j} (1 - 1) mathbf{k} 2 mathbf{k} ]

The surface integral is:

[ iint_{S} ( abla times mathbf{F}) cdot mathbf{n} , dS iint_{S} 2 , dS 2 times text{Area}(S) 2 times pi 2 pi ]

The boundary ( partial S ) is the circle ( x^2 y^2 1 ). Parameterizing this gives ( x cos t, , y sin t ) for ( t in [0, 2pi] ) and ( dmathbf{r} (-sin t, cos t) , dt ).

The line integral is:

[ oint_{partial S} mathbf{F} cdot dmathbf{r} int_0^{2pi} -sin t cos t cdot -sin t cos t , dt int_0^{2pi} 1 , dt 2 pi ]

Thus, by Stokes' Theorem, the integral of the curl of ( mathbf{F} ) over the surface is equal to the line integral around the boundary, both yielding ( 2 pi ).

Summary

Gauss's Theorem connects the divergence of a vector field within a volume to the flux across its boundary surface, making it invaluable for problems involving flux calculations.

Stokes' Theorem connects the curl of a vector field over a surface to the circulation around its boundary, aiding in evaluating circulation and flux more efficiently.

Both theorems are essential tools in physics and engineering, particularly in electromagnetism and fluid dynamics, providing powerful methods to convert complex integrals into simpler calculations.