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The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes

Example

On $U = R^{3} ∖ {0}$ let $F (x, y, z) = \frac{(x, y, z)}{(x^{2} + y^{2} + z^{2})^{3 / 2}} .$ Then $div F = 0$ on $U$ . Consequently the outward flux of $F$ through the sphere bounding the translated unit ball $B = {(x, y, z) : (x^{2} + y^{2} + (z - 2)^{2}) \leq 1}$ is $0$ .

Facts & Assumptions

Given: The field $F$ on $U = R^{3} ∖ {0}$ , and the translated closed unit ball $B = {(x, y, z) : (x^{2} + y^{2} + (z - 2)^{2}) \leq 1}$ .

[F1]

The divergence of a field is the sum of its coordinate partial derivatives (Divergence and curl of a $C^{1}$ vector field).

[L3]

Products differentiate by the product rule (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L4]

Composites differentiate by the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[L6]

For every real $α$ , the function $s \mapsto s^{α}$ is continuous and differentiable on $(0, \infty)$ , with derivative $α s^{α - 1}$ (Continuity and derivatives of positive-base real powers).

[F2]

The Jacobian matrix records the coordinate partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

[F3]

$∥ x ∥_{2} = \sqrt{⟨ x, x ⟩}$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F4]

Flux is computed against the oriented area vector of a patch (Unit normal fields, orientations, and flux through a regular surface patch).

[F5]

A subset of a metric space is open when every one of its points contains an open metric ball lying in the subset; the Euclidean metric on $R^{3}$ is induced by $∥ \cdot ∥_{2}$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement, Open ball, closed ball and sphere in a metric space, $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it).

[L7]

Each coordinate projection on Euclidean space is $1$ -Lipschitz and therefore continuous; finite sums and products of continuous real-valued maps are continuous, as are their composites (Vector-valued functions $f : A \to R^{m}$ , their limits and continuity, with the dictionary to the metric notions, Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent, Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ ).

[L8]

A regular surface patch may identify parameters only on its boundary; its induced area vector gives the outward flux integral (Regular parametrized surface patches on compact Jordan parameter regions, Unit normal fields, orientations, and flux through a regular surface patch).

[L9]

A continuous function on a closed rectangle is Riemann integrable and has the corresponding iterated integral; the integral of an integrable derivative is its endpoint increment (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable, Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections, The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ ).

[L10]

Sine and cosine have the usual derivatives, $\sin^{2} u + \cos^{2} u = 1$ , $\sin u > 0$ for $0 < u < π$ , cosine is strictly decreasing on $[0, π]$ , and $\cos 0 = 1$ , $\cos π = - 1$ (The derivatives of sine and cosine are cosine and minus sine, Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, Quarter-turn values and shifts by pi/2 and pi).

[L11]

The unit-circle parametrization $θ \mapsto (\cos θ, \sin θ)$ is injective on $[0, 2 π)$ ( $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle); the cross product has its coordinate determinant formula (The cross product in $R^{3}$ ).

Verification

technique · direct

1.1L3L4L6L7F2F3given

Put $s (x, y, z) = x^{2} + y^{2} + z^{2}$ , which is positive and continuous on $U$ by [L7]. The $i$ th component of $F$ is $F_{i} = x_{i} s^{- 3 / 2}$ . By the product and chain rules [L3, L4], the positive-base power rule [L6], and the coordinate interpretation of partial derivatives [F2], every coordinate partial derivative is $\partial_{j} F_{i} = {\begin{cases} s^{- 3 / 2} - 3 x_{i}^{2} s^{- 5 / 2}, & j = i, \\ - 3 x_{i} x_{j} s^{- 5 / 2}, & j \neq i . \end{cases}$ The coordinate projections are continuous by [L7], so $s$ is continuous; because $s > 0$ on $U$ , [L6] and [L7] make every function in the displayed formulas continuous there. Hence $F$ is $C^{1}$ on $U$ .

2.1step 1.1F1

Summing the three diagonal formulas of step 1.1 gives $div F = 3 s^{- 3 / 2} - 3 s s^{- 5 / 2} = 0$ on $U$ by [F1].

2.2step 1.1F3F5given

Every point of $B$ has distance at least $1$ from the origin, so $B \subseteq U$ . The set $U$ is open: if $p \in U$ , then $∥ p ∥_{2} > 0$ and the ball $B (p, ∥ p ∥_{2} / 2)$ cannot contain the deleted origin. Hence $F$ is continuous on a neighbourhood of the sphere.

3.1L8L10L11step 2.2construct

Parametrize $\partial B$ by $ψ (ϕ, θ) = (\sin ϕ \cos θ, \sin ϕ \sin θ, 2 + \cos ϕ)$ on $D = [0, π] \times [0, 2 π]$ . Direct differentiation and the cross-product formula give $ψ_{ϕ} \times ψ_{θ} = \sin ϕ (\sin ϕ \cos θ, \sin ϕ \sin θ, \cos ϕ)$ . On $D^{\circ}$ this is nonzero and points outward. Strict monotonicity of cosine on $[0, π]$ and [L11] make the parametrization injective on its interior; its only repeated boundary images lie at the seam or poles. Thus $ψ$ is one regular patch covering the sphere in the sense of [L8].

4.1F3F4L8L9L10step 3.1

On this patch, $∣ ψ ∣^{2} = 5 + 4 \cos ϕ \geq 1$ and $ψ \cdot (ψ_{ϕ} \times ψ_{θ}) = \sin ϕ (1 + 2 \cos ϕ)$ . Thus [F4] and [L9] give the outward flux as $2 π \int_{0}^{π} (1 + 2 \cos ϕ) \sin ϕ (5 + 4 \cos ϕ)^{- 3 / 2} d ϕ$ .

5.1L3L4L6L9L10step 4.1∎

Put $H (u) = \frac{1}{4} (\sqrt{5 + 4 u} + 3 / \sqrt{5 + 4 u})$ for $- 1 \leq u \leq 1$ . The power and chain rules give $H^{'} (u) = (1 + 2 u) (5 + 4 u)^{- 3 / 2}$ , so the integrand in step 4.1 is $- \frac{d}{d ϕ} H (\cos ϕ)$ . Since $H (1) = H (- 1) = 1$ , the fundamental theorem in [L9] makes the flux $- 2 π [H (\cos ϕ)]_{0}^{π} = 0$ .

Remarks

The translation moves the sphere away from the singular origin. The direct patch calculation proves its zero flux without requiring an elementary-solid presentation of the ball.

Depends on

Used by

FALSE: a field with vanishing divergence has zero outward flux through the boundary of every solid it surrounds False statement

Dependency tree · two levels

133 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. Strang and E. Herman, Calculus Volume 3, Examples 6.78-6.80 (standard reference, not scraped)
J. Feldman, A. Rechnitzer and E. Yeager, CLP-4 Vector Calculus, Example 4.4.8 (standard reference, not scraped)