Let F vector = <x,y,z> and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).

Question

Let

F \vec{\to} r = < x, y, z >

and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).

Answer 1

Step 1
Consider the provided question,
Given,

\vec{F} = ⟨ x, y, z ⟩ = ξ + y j + z k

Let s be a closed surface enclosing some volume V.
Find

\int_{s} \vec{F} \cdot N d s

By gauss divergence theorem,

\int_{s} \vec{F} \cdot N d s = \int_{V} \div \vec{F} d v

Since,

\vec{F} = ξ + y j + z k

div

\vec{F} = \frac{\partial}{\partial x} (x) + (\partial y) (y) + (\partial z) (z)

=1+1+1
=3
Step 2
Now, find the (nonzero) volume V of some solid in

R^{3}

by calculating a surface integral.

\int_{s} \vec{F} \cdot N d s = \int_{V} \div \vec{F} d v

= \int_{V} (3) d v

= 3 \int_{V} 1 d v

\int_{s} \vec{F} \cdot N d s = 3 V

V = 1.3 \int_{s} \vec{F} \cdot N d s

Hence, nonzero volume,

V = \frac{1}{3} \int_{s} \vec{F} \cdot N d s

Want to know more about Green's, Stokes', and the divergence theorem?