Vector FieldsHard

Divergence theorem

Calculate the flux of F=(x,y,z) through the sphere x²+y²+z²=R² with R=2.

Given data

F = (x,y,z)Sphere r=R=2

Steps

0 / 3

Calculate div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
Write your working — the numerical result must appear in the text
Calculate the volume of the sphere with R=2.
Apply the divergence theorem: Φ = ∭div F dV.

Full worked solution

Calculate div F = ∂P/∂x + ∂Q/∂y + ∂R/∂z.
$\nabla\cdot\mathbf{F} = 1+1+1 = 3$
$\operatorname{div} \mathbf{F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z) = 1 + 1 + 1 = \mathbf{3}$ . The divergence of the radial field $(x,y,z)$ is constant.
Calculate the volume of the sphere with R=2.
$V = \frac{4}{3}\pi(2)^3 = \frac{32\pi}{3}$
$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi \cdot 2^3 = \frac{4}{3}\pi \cdot 8 = \frac{32\pi}{3} \approx \mathbf{33.51}$ . The volume of a sphere of radius $R=2$ .
Apply the divergence theorem: Φ = ∭div F dV.
$\Phi = 3\cdot\frac{32\pi}{3} = 32\pi \approx 100.5$
$\Phi = \iiint_V \operatorname{div} \mathbf{F}\,dV = 3 \times \frac{32\pi}{3} = \mathbf{32\pi} \approx \mathbf{100.5}$ . The flux through the closed surface equals the volume integral of the divergence.

Result:Φ = 32π ≈ 100.5.

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