Explanation Divergence Theorem says about the relation between surface and volume integral. Let S be a closed smooth surface, which is boundary of V, with normal pointing outwards. Let F = 〈 x , y , z 〉 on a region containing V, then the flux is equal to triple integral of div F over the solid region V. ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Given: F ( x , y , z ) = 〈 x 3 , 0 , z 3 〉 , S is the boundary of the region in the first octant of space given by x 2 + y 2 + z 2 ≤ 4 , and x ≥ 0 , y ≥ 0 , z ≥ 0 . Formula Used: ∬ S F ⋅ d S = ∭ V d i v ( F ) d V Calculation: First we have to calculate the div F and we get: d i v ( F ) = ∂ ∂ x ( x 3 ) + ∂ ∂ y ( 0 ) + ∂ ∂ z ( z 3 ) = 3 x 2 + 0 + 3 z 2 = 3 ( x 2 + z 2 ) ∬ S F ⋅ d S = ∭ V 3 ( x 2 + z 2 ) d x d y d z Now, change the variables into spherical coordinates and we get: x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ and d V = ρ 2 sin ϕ d ρ d θ d ϕ So, the triple integral will become: ∭ V 3 ( x 2 + z 2 ) ρ 2 sin ϕ d ρ d θ d ϕ Now, the range of ρ , θ and ϕ is: 0 ≤ ρ ≤ 2 , 0 ≤ θ ≤ π 2 and 0 ≤ ϕ ≤ π Now, calculate x 2 + z 2 and we get: x 2 + z 2 = ρ 2 sin 2 ϕ cos 2 θ + ρ 2 cos 2 ϕ = ρ 2 ( sin 2 ϕ ( 1 − sin 2 θ ) + cos 2 ϕ ) = ρ 2 ( sin 2 ϕ + cos 2 ϕ − sin 2 ϕ sin 2 θ ) = ρ 2 ( 1 − sin 2 ϕ sin 2 θ ) Now, calculate the triple integral and we get:

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ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 18.3, Problem 11E

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The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

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Chapter 18.3, Problem 10E

Chapter 18.3, Problem 12E

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Use the Divergence Theorem to compute the flux of F across the surface of the solid S: F(x, y, z) = ⟨x^3 , y^3 , z^3 ⟩ and S is the solid bounded by z = 0 and 4 = x^2 + y^2 + z^2 .

Use Divergence theorem to find the outward flux of F = 2xz i - 2xy j – z2 k across the boundary of the region cut from the first octant by the plane y + z = 4 and the elliptical cylinder 4x2 + y2 = 16.

Use the Divergence Theorem to find the flux of F = xy2i + x2yj + yk outward through the surface of the region enclosed by the cylinder x2 + y2 = 1 and the planes z = 1 and z =-1.

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The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, with normal pointing outwards.

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The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.

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Chapter 18 Solutions

The flux ∬ S F ⋅ d S by using the Divergence Theorem.

Solution Summary: The author calculates the flux by using the Divergence Theorem. Let S be a closed smooth surface, with normal pointing outwards.

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To calculate: The flux ∬SF⋅dS by using the Divergence Theorem.

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Chapter 18 Solutions

To calculate:

The flux $\iint_{S} F \cdot d S$ by using the Divergence Theorem.