Explain why or why not Determine whether the following statementsare true and give an explanation or counterexample.a. If ∇ ⋅ F = 0 at all points of a region D, then F ⋅ n = 0 at allpoints 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.
Explain why or why not Determine whether the following statementsare true and give an explanation or counterexample.a. If ∇ ⋅ F = 0 at all points of a region D, then F ⋅ n = 0 at allpoints 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.
Elementary Linear Algebra (MindTap Course List)
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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
a.
If at all points of a region D, then by divergence theorem,
This implies that at all points of the boundary points of D.
So, the statement is "TRUE".
Step 2
b.
If at all points of a region D, then by divergence theorem,
So, this means,
So, it means F is constant.
So, the statement is "TRUE".
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