Answered: Explain why or why not Determine…

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

Explain why or why not Determine whether the following statements
are true and give an explanation or counterexample.
a. If ∇ ⋅ F = 0 at all points of a region D, then F ⋅ n = 0 at all
points of the boundary of D.
b. If ∫∫_S F ⋅ n dS = 0 on all closed surfaces in ℝ³, then F is constant.
c. If | F| < 1, then | ∫∫∫_D ∇ ⋅ F dV | is less than the area of the surface of D.

Expert Solution

Step 1

If $\nabla . F = 0$ at all points of a region D, then by divergence theorem,

$\int \int_{S} F . n d S = \int \int \int_{V} \nabla . F d V \int \int_{S} F . n d S = 0$

This implies that $F . n = 0$ at all points of the boundary points of D.

So, the statement is "TRUE".

Step 2

If $F . n = 0$ at all points of a region D, then by divergence theorem,

$\int \int_{S} F . n d S = \int \int \int_{V} \nabla . F d V \int \int \int_{V} \nabla . F d V = 0$

So, this means,

$\nabla . F = 0$

So, it means F is constant.

So, the statement is "TRUE".

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