Respuesta :
Answer:
36π
Explanation:
Close the surface (call it S) by including the cylindrical caps using the given planes.
Now, we can apply the Divergence Theorem:
∫∫s F · dS
= ∫∫∫ div F dV
= ∫∫∫ (1+1+1) dV
= ∫(r = 0 to 3) ∫(θ = 0 to 2π) ∫(z = 1 - r cos θ - r sin θ to 5 - r cos θ - r sin θ) 3 dz dθ dr, via cylindrical coordinates
= ∫(r = 0 to 3) ∫(θ = 0 to 2π) 12 dθ dr
= 72π.
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If the surface was not originally closed, we need to subtract the flux contributions from the caps.
(i) Bottom cap via z = 1 - x - y (with downward pointing normal)
∫∫s₁ F · dS
= ∫∫ <x, y, z> · -<-z_x, -z_y, 1> dA
= ∫∫ <x, y, 1 - x - y> · <-1, -1, -1> dA
= ∫∫ -1 dA
= -9π, being the area inside x^2 + y^2 = 9.
(ii) Upper cap via z = 5 - x - y (with upward pointing normal)
∫∫s₁ F · dS
= ∫∫ <x, y, z> · <-z_x, -z_y, 1> dA
= ∫∫ <x, y, 5 - x - y> · <1, 1, 1> dA
= ∫∫ 5 dA
= 5 * 9π
= 45π.
Hence, the flux for the capless surface equals
72π - (-9π) - 45π = 36π.
I hope this helps!
The flux under the given conditions will be 36π
What is Flux?
This is defined as a measurement of the total magnetic field which passes through a given area.
Using Divergence Theorem,
∫∫s F · dS = ∫∫∫ div F dV
= ∫∫∫ (1+1+1) dV
= ∫(r = 0 to 3) ∫(θ = 0 to 2π) ∫(z = 1 - r cos θ - r sin θ to 5 - r cos θ - r sin θ) 3 dz dθ dr, via cylindrical coordinates
= ∫(r = 0 to 3) ∫(θ = 0 to 2π) 12 dθ dr
= 72π.
Surface was not originally closed so we subtract the flux contributions from the caps.
- Bottom cap via z = 1 - x - y (with downward pointing normal)
∫∫s₁ F · dS = ∫∫ <x, y, z> · -<-z_x, -z_y, 1> dA
= ∫∫ <x, y, 1 - x - y> · <-1, -1, -1> dA
= ∫∫ -1 dA
= -9π
- Upper cap via z = 5 - x - y (with upward pointing normal)
∫∫s₁ F · dS = ∫∫ <x, y, z> · <-z_x, -z_y, 1> dA
= ∫∫ <x, y, 5 - x - y> · <1, 1, 1> dA
= ∫∫ 5 dA
= 5 * 9π
= 45π.
Surface flux = 72π - (-9π) - 45π
= 36π.
Read more about Flux here https://brainly.com/question/26289097
