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 boundary 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.

Step-by-step explanation:

1) True. This is because the divergence of F is 1, thus, F is a linear function. Orientation is given outward to the surface. Linear function double integrated over a surface with outward orientation gives volume enclosed by the surface.

2) True. This is primarily what the Divergence theorem is.

3) False. If F was 3/pi instead of div (F), then the statement would have been true.

4) False. The gradient of divergence can be anything. The curl of divergence of a vector function is 0, not the gradient o divergence.

5) False. While finding Divergence, derivatives are taken for different variables. Since the derivatives of constants are 0, therefore, both the vector functions F and G can be different constant parts of there components even if their divergences are equal.