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

Example 17 :
A vector field is given by $\vec F = 4xz\hat\imath -y^2\hat\jmath + yz\hat k$. Find the surface integral of the field from the surfaces of a unit cube bounded by planes $x=0, x=1, y=0, y=1, z=0$and $z=1$. Verify that the result agrees with the divergence theorem.

Solution :
Divergence of $\vec F$is

\begin{eqnarray*} \nabla\cdot\vec F &=& \frac{\partial F_x}{\partial x}+ \frac{\... ...\partial y} + \frac{\partial yz}{\partial z}\\ &=& 4z-2y+y=4z-y \end{eqnarray*}

The volume integral of above is

\begin{displaymath}\int\nabla\cdot\vec F dV = \int_0^1 dx\int_0^1 dy\int_0^1 (4z-y)dxdydz= \frac{3}{2}\end{displaymath}

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