When To Use Divergence Theorem

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Section 6-6 : Divergence Theorem

In this department we are going to chronicle surface integrals to triple integrals. Nosotros will do this with the Divergence Theorem.

Divergence Theorem

Allow \(East\) be a elementary solid region and \(Due south\) is the boundary surface of \(Eastward\) with positive orientation. Allow \(\vec F\) be a vector field whose components take continuous starting time social club partial derivatives. Then,

\[\iint\limits_{Southward}{{\vec F\centerdot d\vec S}} = \iiint\limits_{E}{{{\mathop{\rm div}\nolimits} \vec F\,dV}}\]

Let'due south run across an instance of how to utilise this theorem.

Example 1 Use the difference theorem to evaluate \(\displaystyle \iint\limits_{Southward}{{\vec F\centerdot d\vec S}}\) where \(\vec F = xy\,\vec i - \frac{1}{2}{y^ii}\,\vec j + z\,\vec k\) and the surface consists of the iii surfaces, \(z = four - 3{x^2} - 3{y^2}\), \(1 \le z \le 4\) on the top, \({ten^ii} + {y^2} = 1\), \(0 \le z \le 1\) on the sides and \(z = 0\) on the bottom.

Show Solution

Let's start this off with a sketch of the surface.

This is a graph with the standard 3D coordinate system. The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward. The walls of the solid in this graph are the cylinder given in the problem statement and whose

The region \(E\) for the triple integral is then the region enclosed by these surfaces. Note that cylindrical coordinates would be a perfect coordinate system for this region. If we practice that here are the limits for the ranges.

\[\brainstorm{assortment}{c}0 \le z \le iv - three{r^2}\\ 0 \le r \le 1\\ 0 \le \theta \le ii\pi \end{assortment}\]

We'll likewise need the deviation of the vector field so permit'south go that.

\[{\mathop{\rm div}\nolimits} \vec F = y - y + 1 = 1\]

The integral is then,

\[\begin{align*}\iint\limits_{S}{{\vec F\centerdot d\vec S}} & = \iiint\limits_{Eastward}{{{\mathop{\rm div}\nolimits} \vec F\,dV}}\\ & = \int_{{\,0}}^{{\,two\pi }}{{\int_{{\,0}}^{{\,ane}}{{\int_{{\,0}}^{{4 - 3{r^ii}}}{{r\,dz}}\,dr}}\,d\theta }}\\ & = \int_{{\,0}}^{{\,ii\pi }}{{\int_{{\,0}}^{{\,1}}{{4r - three{r^3}\,dr}}\,d\theta }}\\ & = \int_{{\,0}}^{{\,2\pi }}{{\left. {\left( {2{r^2} - \frac{iii}{4}{r^4}} \correct)} \right|_0^i\,d\theta }}\\ & = \int_{{\,0}}^{{\,ii\pi }}{{\frac{5}{4}\,d\theta }}\\ & = \frac{5}{2}\pi \end{align*}\]