Solution Found!
Explain why or why not Determine whether the
Chapter 14, Problem 31E(choose chapter or problem)
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If \(\nabla \cdot \mathbf{F}=0\) at all points of a region D. then \(\mathbf{F} \cdot \mathbf{n}=0\) at all points of the boundary of D.
b. If .\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=0\) on all closed surfaces in \(\mathbf{R}^{3}\), then F is constant.
c. If |F| < l. then \(\left|\iiint_{D} \nabla \cdot \mathbf{F} d V\right|\) dvl is less than the area of the surface of D.
Text Transcription:
Nabla cdot F = 0
F cdot = 0
iint_SF cdot ndS = 0
|iiint_D nabla cdot F dV|
Questions & Answers
QUESTION:
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If \(\nabla \cdot \mathbf{F}=0\) at all points of a region D. then \(\mathbf{F} \cdot \mathbf{n}=0\) at all points of the boundary of D.
b. If .\(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=0\) on all closed surfaces in \(\mathbf{R}^{3}\), then F is constant.
c. If |F| < l. then \(\left|\iiint_{D} \nabla \cdot \mathbf{F} d V\right|\) dvl is less than the area of the surface of D.
Text Transcription:
Nabla cdot F = 0
F cdot = 0
iint_SF cdot ndS = 0
|iiint_D nabla cdot F dV|
ANSWER:Solution 31E(a) The given statement “If F = 0 at all points of a region then Fn = 0 all points of the boundary of ” is false.For example F = has F at all points of the unit sphere, but the normal