Module 1 : A Crash Course in Vectors
Lecture 4 : Gradient of a Scalar Function

Example 20 :

A hemispherical bowl of radius 1 lies with its base on the x-y plane and the origin at the centre of the circular base. Calculate the surface integral of the vector field $\vec F = x^3\hat\imath + y^3\hat\jmath + z^3\hat k$in the hemisphere and verify the divergence theorem.

Solution :
The divergence of $\vec F$is easily calculated

\begin{eqnarray*} \nabla\cdot\vec F &=& \frac{\partial F_x}{\partial x}+ \frac{\... ...y} + \frac{\partial z^3}{\partial z}\\ &=& 3(x^2+y^2+z^2)= 3r^2 \end{eqnarray*}

where $r$is the distance from origin. The volume integral over the hemisphere is conveniently calculated in spherical polar using the violume element $r^2\sin\theta d\theta d\phi$. Since it is a hemisphere with $z=0$as the base, the range of $\theta$is $0$to $\pi /2$.

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