Verification of Gauss's Divergence Theorem for the Vector Field ( mathbf{A} xmathbf{i} ymathbf{j} zmathbf{k} ) on the Surface of a Sphere

In this article, we will verify Gauss's Divergence Theorem for the vector field ( mathbf{A} x mathbf{i} y mathbf{j} z mathbf{k} ) over the surface of a sphere defined by ( x^2 y^2 z^2 a^2 ).

Step 1: Compute the Divergence of ( mathbf{A} )

The divergence of a vector field ( mathbf{A} P mathbf{i} Q mathbf{j} R mathbf{k} ) is given by:

( abla cdot mathbf{A} frac{partial P}{partial x} frac{partial Q}{partial y} frac{partial R}{partial z} )

For the given vector field, ( P x ), ( Q y ), and ( R z ).

Calculating the divergence:

( abla cdot mathbf{A} frac{partial x}{partial x} frac{partial y}{partial y} frac{partial z}{partial z} 1 1 1 3 )

Step 2: Compute the Volume Integral of the Divergence

The volume ( V ) enclosed by the sphere is given by the formula:

( V frac{4}{3} pi a^3 )

The volume integral of the divergence over ( V ) is:

( iiint_V abla cdot mathbf{A} , dV iiint_V 3 , dV )

The integral simplifies to:

( iiint_V 3 , dV 3 cdot frac{4}{3} pi a^3 4 pi a^3 )

Step 3: Compute the Surface Integral of ( mathbf{A} ) over the Sphere

The surface integral of ( mathbf{A} ) over the sphere ( S ) is given by:

( iint_S mathbf{A} cdot dmathbf{S} )

Where ( dmathbf{S} hat{n} , dS ) and ( hat{n} ) is the outward normal vector. For the sphere, ( hat{n} frac{mathbf{r}}{a} frac{x mathbf{i} y mathbf{j} z mathbf{k}}{a} ).

The differential surface area element ( dS ) on the sphere can be directly computed as:

( mathbf{A} cdot dmathbf{S} mathbf{A} cdot hat{n} , dS (x mathbf{i} y mathbf{j} z mathbf{k}) cdot left(frac{x mathbf{i} y mathbf{j} z mathbf{k}}{a}right) a^2 , dS )

On the surface of the sphere, ( x^2 y^2 z^2 a^2 ). Thus:

( mathbf{A} cdot hat{n} frac{x^2 y^2 z^2}{a} frac{a^2}{a} a )

Therefore:

( iint_S mathbf{A} cdot dmathbf{S} iint_S a , dS a iint_S dS )

The total surface area of the sphere is:

( text{Surface Area} 4 pi a^2 )

Hence:

( iint_S mathbf{A} cdot dmathbf{S} a cdot 4 pi a^2 4 pi a^3 )

Conclusion

We have verified that the surface integral and the volume integral yield the same result:

( iint_S mathbf{A} cdot dmathbf{S} 4 pi a^3 )

( iiint_V abla cdot mathbf{A} , dV 4 pi a^3 )

Thus, we have verified Gauss's Divergence Theorem for the vector field ( mathbf{A} x mathbf{i} y mathbf{j} z mathbf{k} ) over the surface of the sphere.