Let S be the square in the xy-plane, oriented with the normal pointing in the positive z-direction. Estimate ∬S F⋅dS where F is a vector field whose values at the labeled points are F(A)=⟨9,3,5⟩, F(C)=⟨9,8,−4⟩, F(B)=⟨−9,8,−7⟩, F(D)=⟨−3,4,−1⟩.

I disliked the above answer solely because this is a WebWork problem that requires an input, and "not enough information" is not a valid answer...

To estimate ∬S F⋅dS, we can use the divergence theorem, which states that the flux of a vector field F across a closed surface S is equal to the triple integral of the divergence of F over the region V enclosed by S.

To calculate the flux, we need to find the divergence of F and then integrate it over the region enclosed by S.

Step 1: Find the divergence of F
The divergence of F is given by
div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z

Given:
F(A) = ⟨9,3,5⟩
F(C) = ⟨9,8,−4⟩
F(B) = ⟨−9,8,−7⟩
F(D) = ⟨−3,4,−1⟩

∂Fx/∂x = 0 (since Fx is constant over the region)
∂Fy/∂y = 0 (since Fy is constant over the region)
∂Fz/∂z = 0 (since Fz is constant over the region)

Therefore, div(F) = 0 + 0 + 0 = 0

Step 2: Integrate the divergence over the region enclosed by S
Since the divergence of F is zero, ∬S F⋅dS = 0.

Therefore, the estimated value of ∬S F⋅dS is 0.

To estimate ∬S F⋅dS, we first need to calculate the surface area of the square S in the xy-plane. Let's denote the side length of the square as "s".

The surface area of a square is calculated by squaring the side length. Therefore, the surface area of S is given by A = s^2.

Next, we need to calculate the normal vector to the surface of the square. Since the square is oriented with the normal pointing in the positive z-direction, the normal vector can be represented as N = ⟨0, 0, 1⟩.

Now, we can calculate the integral ∬S F⋅dS by summing up the dot product of the vector field F and the normal vector N at each labeled point on the square, and then multiplying it by the surface area.

∬S F⋅dS = (F(A)⋅N + F(B)⋅N + F(C)⋅N + F(D)⋅N) * A

= ((9, 3, 5)⋅⟨0, 0, 1⟩ + (-9, 8, -7)⋅⟨0, 0, 1⟩ + (9, 8, -4)⋅⟨0, 0, 1⟩ + (-3, 4, -1)⋅⟨0, 0, 1⟩) * A

= (0 + 0 + 0 + 0) * A

= 0 * A

= 0

Therefore, the estimated value of ∬S F⋅dS is 0.

Usually, this just the limit of the sum

∑F⋅n ∆S
but you have only provided a few discrete values of F, and have not indicated how S is to be subdivided.

Maybe you can review surface integrals and provide a bit more detail...