Are the following statements true or false? 1. If F⃗ is a vector field in 3-dimensional space, and W is a solid region with boun

Question

19 days ago

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Are the following statements true or false? 1. If F⃗ is a vector field in 3-dimensional space, and W is a solid region with boun

dary surface S, then ∬SF⃗ ⋅dS⃗ =∭Wdiv(F⃗ )dV. 2. If F⃗ and G⃗ are vector fields satisfying div(F⃗ )=div(G⃗ ), then F⃗ =G⃗ . 3. If ∬SF⃗ ⋅dS⃗ =12 and S is a flat disk of area 4π, then div(F⃗ )=3/π. 4. If F⃗ is a vector field in 3-dimensional space satisfying div(F⃗ )=1, and S is a closed surface oriented outward, then ∬SF⃗ ⋅dS⃗ is equal to the volume enclosed by S. 5. If F⃗ is a vector field in 3-dimensional space, then grad(div(F⃗ ))=0⃗ .

Mathematics

1 answer:

lawyer [9.2K] · Answer 1 · 2026-03-08T12:10:29+03:00

Answer:

Step-by-step explanation:

1) True. This stems from the fact that the divergence of F is 1, indicating that F is a linear function. The orientation is outward from the surface. Integrating a linear function over a surface with outward orientation leads to the volume enclosed by that surface.

2) True. This is fundamentally what the Divergence theorem states.

3) False. Had F been specified as 3/pi instead of div(F), this claim would have held true.

4) False. The gradient of divergence can vary. The curl of the divergence of a vector function is 0, contradicting the notion of the gradient of divergence being 0.

5) False. While calculating divergence, derivatives are computed for different variables. Since the derivative of constants is 0, both vector functions F and G can contain distinct constant components even when their divergences are equal.