Exercise 1518Calculus › Multivariable calculus › Vector calculusL3Added 2 Nov 2026

Divergence theorem (Ostrogradsky)

#divergence theorem#flux#surface

Statement

Let $S$ be the sphere of radius $R$ centered at the origin, oriented outward. Compute the flux of the field $F(x,y,z) = (x,y,z)$ through $S$ .

Hint

Apply the divergence theorem: $\iint_S F \cdot n \, dS = \iiint_V \operatorname{div} F \, dV$ .

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Solution

Divergence.
$\operatorname{div} F = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 3.$

Divergence theorem.
$\iint_S F \cdot n \, dS = \iiint_V 3 \, dV = 3 \cdot \frac43 \pi R^3 = 4\pi R^3.$

Remark. This result is also the product of the area of the sphere ( $4\pi R^2$ ) and the mean normal component of the field ( $R$ ), giving $4\pi R^3$ .

Divergence theorem (Ostrogradsky)

Statement

Solution

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