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.

the solution of the equation is 2.23 ⇒ answer b

step-by-step explanation:

* lets revise the meaning of exponential function

- the form of the exponential function is y = ab^x, where a ≠ 0, b > 0 ,  

  b ≠ 1, and x is any real number

- it has a constant base b

- it has a variable exponent x

- to solve an exponential equation, take the log or ln of both sides,  

  and solve for the variable

* lets solve the problem

∵

- the base is 7 and the exponent is x

- insert ㏑ in both sides

∴

- use the rule

∴ x ㏑(7) = ㏑(77)

- to find x divide both sides by ㏑(7)

∴

* the solution of the equation is 2.23 to the nearest 2 decimal places

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step-by-step explanation:

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