Consider a region D enclosed by the surface S with outward normal n. Use the divergencetheorem to show that the volume of D is given by1/ / / / ²3Where r is the position vector (x, y, z) of a point.V =r.ndo

Answered: Consider a region D enclosed by the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Consider a region D enclosed by the surface S with outward normal n. Use the divergence
theorem to show that the volume of D is given by
1/ / / / ²
3
Where r is the position vector (x, y, z) of a point.
V =
r.ndo

Expert Solution

Step 1

What is Divergence Theorem:

According to the divergence theorem, the volume integral of the divergence over the area inside the surface equals the surface integral of a vector field over a closed surface, also known as the flux through the surface. It implies that the net flux out of a region is equal to the sum of all the field sources in that region. A crucial finding in the mathematics of physics and engineering, particularly in electrostatics and fluid dynamics, is the divergence theorem. It is typically used in three dimensions in these fields.

Given:

Given region $D$ is enclosed by the surface $S$ with outward normal $n$ . Also, it is provided that the position vector of a point be, $r = x i + y j + z k$ .