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Transport of flux through a moving patch

Statement

Assume the Axiom of Countable Choice ( ${A C}_{ω}$ , The Axiom of Countable Choice ( ${A C}_{ω}$ )). In the setting of Moving parametrized surface patches and their flux: $D \subseteq R^{2}$ is a finite elementary Green region with positive boundary chain $\partial D$ , $I \subseteq R$ is an interval, $r : D \times I \to R^{3}$ extends to a $C^{2}$ map on a neighbourhood of $D \times I$ with $r_{u} \times r_{v} \neq 0$ on $D^{\circ} \times I^{\circ}$ , $(W, v)$ is a $C^{1}$ velocity field with $v (r (p, t), t) = r_{t} (p, t)$ , and $B$ is a $C^{1}$ field on an open set containing the swept graph $Σ = {(r (p, t), t) : p \in D, t \in I} \subseteq R^{4}$ , with flux

$Φ (t) = \int_{D} ⟨ B (r (p, t), t), r_{u} (p, t) \times r_{v} (p, t) ⟩ d p .$

For nondegenerate $I$ , $Φ$ is differentiable at interior times and relatively differentiable at included endpoints. Empty $I$ gives no flux values, and singleton $I$ gives one flux value but no derivative determined by that value. If the spatial divergence $div B = 0$ , then for every time of such a nondegenerate $I$ whose slice has rank two, that is $r_{u} (p, t) \times r_{v} (p, t) \neq 0$ for all $p \in D^{\circ}$ ,

$Φ^{'} (t) = \int_{S (t)} \partial_{t} B \cdot n d S - \oint_{\partial S (t)} (v \times B) \cdot d r .$

By the standing hypothesis the slice has rank two at every interior time. At an included endpoint the displayed normal notation is used only if its slice also has rank two. Here $S (t)$ means the parametrized family $(D, r (\cdot, t))$ , and $n$ means the parameter normal $N (p, t) = (r_{u} \times r_{v}) / ∥ r_{u} \times r_{v} ∥_{2}$ on $D^{\circ}$ . Define $\int_{S (t)} H \cdot n d S : = \int_{D} ⟨ H (r (p, t), t), r_{u} \times r_{v} ⟩ d p$ , including multiplicity; similarly $\int_{S (t)} (v \cdot n) div B d S : = \int_{D} ⟨ v (r (p, t), t), r_{u} \times r_{v} ⟩ div B (r (p, t), t) d p$ . These agree with the ordinary flux definitions when the slice meets Regular parametrized surface patches on compact Jordan parameter regions (through Moving parametrized surface patches and their flux); no image-wide normal or injectivity is assumed. If $\partial D = (σ_{1}, \dots, σ_{m})$ , define $\oint_{\partial S (t)} G \cdot d r : = \sum_{ℓ = 1}^{m} \int_{r (σ_{ℓ}, t)} G (\cdot, t) \cdot d r$ using Positive orientation of elementary-region boundaries and Scalar line integrals with respect to arc length and vector-field line integrals. Without the hypothesis $div B = 0$ the additional term

$\int_{S (t)} (v \cdot n) div B d S$

appears on the right-hand side. The identity is local in $t$ and is a statement about the parametrized mathematical object; no electromagnetic interpretation is asserted.

Facts & Assumptions

Given: The data $(D, I, r, v, B, Φ)$ of the Statement, a nondegenerate $I$ and times $t \in I$ that are interior or whose slice has rank two (the two cases needed below), and the notation $w : = r_{u} \times r_{v}$ , $N : = w / ∥ w ∥_{2}$ on $D^{\circ}$ at a rank-two slice, and $\tilde{v} (p, t) : = r_{t} (p, t) = v (r (p, t), t)$ .

[F1]

Moving patch: the flux is the integral displayed in the Statement over the compact Jordan measurable region $D$ ; the velocity field is $C^{1}$ on an open spacetime neighbourhood of the swept graph and satisfies $v (r (p, t), t) = r_{t} (p, t)$ ; $r$ extends $C^{2}$ to a neighbourhood of $D \times I$ (Moving parametrized surface patches and their flux).

[F2]

Differentiation under the integral sign over the compact measurable set $D$ : if a real-valued function and its partial $t$ -derivative are continuous on a neighbourhood of $D \times [a, b]$ , then $t \mapsto \int_{D} f (p, t) d p$ is differentiable with the stated relative endpoint derivatives and derivative $\int_{D} \partial_{t} f (p, t) d p$ (Differentiation under the integral sign over a compact measurable set and over a compact $C^{1}$ surface). To apply its Lebesgue formulation to this Riemann flux, use Borel Darboux integrands in finite dimension on the bounded Borel zero extension from the compact Jordan set $D$ to a bounding box; it is Riemann integrable by the flux definition and the same is true of the continuous derivative integrand.

[F3]

Calculus rules: chain rule for total derivatives, the product rule, equality of mixed partials for $C^{2}$ maps, and the $C^{k}$ conventions (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , 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 $g (c) \neq 0$ , Clairaut--Schwarz theorem for continuous second partial derivatives, $C^{k}$ Euclidean maps and diffeomorphisms).

[F4]

Cross product and mixed product: $u \times v$ is bilinear and alternating and $⟨ u \times v, w ⟩ = \det [u v w]$ , so the scalar triple product is multilinear and alternating in its three arguments (The cross product in $R^{3}$ , The cross product is bilinear, alternating, and orthogonal to both factors); the Euclidean inner product is that of The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

[F5]

Rigidity of alternating column-multilinear matrix functions: if $F : M_{3} (R) \to R$ is column-multilinear in the sense of Column-multilinear, alternating, normalized and antisymmetric functions on square matrices over a commutative ring and alternating (it vanishes whenever two columns are equal), then $F (M) = F (I_{3}) \det (M)$ for every $M$ ; in particular a column-multilinear alternating $F$ with $F (I_{3}) = 0$ vanishes identically (Every alternating multilinear $F$ satisfies $F (A) = F (I) \sum_{σ \in S_{n}} sgn (σ) \prod_{i} a_{σ (i), i}$ ).

[F6]

Cross-product differential identity: for $C^{1}$ fields $F, G$ , $curl (F \times G) = (div G) F - (div F) G + D F G - D G F$ with divergence and curl of Divergence and curl of a $C^{1}$ vector field (The divergence and curl of a cross product).

[F7]

Parametric Stokes prerequisites, with no rank or injectivity hypothesis: The curl flux integrand of a $C^{2}$ patch is a two-dimensional curl of the pulled-back field gives $\partial_{u} Q^{*} - \partial_{v} P^{*} = ⟨ curl G \circ φ, φ_{u} \times φ_{v} ⟩$ for $P^{*} = ⟨ G \circ φ, φ_{u} ⟩$ , $Q^{*} = ⟨ G \circ φ, φ_{v} ⟩$ on an open parameter neighbourhood. Green's theorem for finite unions of elementary regions integrates this over any supplied finite elementary Green region, and A vector line integral along an image arc is the parameter line integral of the pulled-back field identifies each resulting boundary arc integral with its image line integral.

[F8]

Surface integrals: for a regular patch, $\int_{S} q d S = \int_{D} (q \circ φ) J_{φ}$ with area density $J_{φ} = ∥ φ_{u} \times φ_{v} ∥_{2}$ , and the flux in the induced orientation is $\int_{D} ⟨ F \circ φ, φ_{u} \times φ_{v} ⟩$ (Surface area and scalar surface integrals on a regular patch, The surface area density is the norm of the cross product of the parameter tangents, Unit normal fields, orientations, and flux through a regular surface patch).

Proof

technique · direct; differentiate the parametrized flux under the integral sign, express the derivative of the area vector by the determinant identity, convert the resulting curl term into a boundary circulation by Stokes, and identify each term with the corresponding surface integral

1.1givenF1F3algebra

Preliminaries. The integrand $h (p, t) : = ⟨ B (r (p, t), t), w (p, t) ⟩$ is defined for $(p, t)$ in a neighbourhood of $D \times I$ and is continuous there together with $\partial_{t} h$ : $r$ and its first derivatives are continuous, $B$ is $C^{1}$ on an open spacetime neighbourhood of the swept graph, and sums, products and compositions of continuous maps are continuous. Moreover $\tilde{v} (p, t) = r_{t} (p, t) = v (r (p, t), t)$ is $C^{1}$ in $(p, t)$ , so $D_{x} v$ denotes the spatial Jacobian of the velocity field, evaluated at $r (p, t)$ , and $div v$ its spatial divergence; at an interior time or a rank-two endpoint slice the vector $w = r_{u} \times r_{v}$ does not vanish on $D^{\circ}$ , so the unit normal $N = w / ∥ w ∥_{2}$ is defined there.

1.2givenF4F5algebra

Algebraic identity for the area vector. For every $3 \times 3$ real matrix $A$ and all $B_{0}, u_{1}, u_{2} \in R^{3}$ , $⟨ B_{0}, (A u_{1}) \times u_{2} ⟩ + ⟨ B_{0}, u_{1} \times (A u_{2}) ⟩ = tr (A) ⟨ B_{0}, u_{1} \times u_{2} ⟩ - ⟨ A B_{0}, u_{1} \times u_{2} ⟩$ . To prove it, let $M = [m_{1} m_{2} m_{3}]$ range over $3 \times 3$ matrices and put $E (M) : = ⟨ m_{1}, (A m_{2}) \times m_{3} ⟩ + ⟨ m_{1}, m_{2} \times (A m_{3}) ⟩ - tr (A) ⟨ m_{1}, m_{2} \times m_{3} ⟩ + ⟨ A m_{1}, m_{2} \times m_{3} ⟩$ . Each of the four terms is linear in each column separately: the triple product is trilinear by [F4] and multiplication by the fixed matrix $A$ is linear, so $E$ is column-multilinear. Writing the four terms as the determinants $T_{1} = \det [m_{1}, A m_{2}, m_{3}]$ , $T_{2} = \det [m_{1}, m_{2}, A m_{3}]$ , $T_{3} = - tr (A) \det [m_{1}, m_{2}, m_{3}]$ and $T_{4} = \det [A m_{1}, m_{2}, m_{3}]$ by [F4], each interchange of two of the columns $m_{1}, m_{2}, m_{3}$ carries every $T_{i}$ to $\pm$ one of the four terms in such a way that the total reverses sign: for example exchanging the first two columns sends $T_{1}$ to $- T_{4}$ , $T_{4}$ to $- T_{1}$ , and $T_{2}, T_{3}$ to their own negatives, so $E \mapsto - E$ ; the other two interchanges are verified by the same determinant expansion. Hence $E$ is alternating as well. Therefore [F5] gives $E (M) = E (I_{3}) \det (M)$ for every $M$ , and $E (I_{3}) = A_{22} + A_{33} - tr (A) + A_{11} = 0$ , because $⟨ e_{1}, A e_{2} \times e_{3} ⟩ = A_{22}$ , $⟨ e_{1}, e_{2} \times A e_{3} ⟩ = A_{33}$ and $⟨ A e_{1}, e_{2} \times e_{3} ⟩ = A_{11}$ computed from the coordinate formula of [F4]. Hence $E \equiv 0$ , which is the displayed identity at $(B_{0}, u_{1}, u_{2}) = (m_{1}, m_{2}, m_{3})$ .

2.1step 1.1F2algebra

Differentiation of the flux. Since $h$ and $\partial_{t} h$ are continuous on a neighbourhood of $D \times I$ , use [F2] on any local compact time segment with $K = D$ ; its Riemann/Lebesgue bridge identifies both parameter integrals and gives that $Φ$ is differentiable on $I$ , with relative one-sided derivatives at included endpoints, and $Φ^{'} (t) = \int_{D} \partial_{t} h (p, t) d p$ .

2.2step 1.2F3F4algebra

The derivative of the integrand. Since $r_{t} = \tilde{v}$ and $\tilde{v} (p, t) = v (r (p, t), t)$ , the chain rule [F3] gives $\partial_{t} r_{u} = \partial_{u} r_{t} = (D_{x} v) r_{u}$ and $\partial_{t} r_{v} = (D_{x} v) r_{v}$ , so, using equality of mixed partials for the $C^{2}$ map $r$ , $\partial_{t} w = ((D_{x} v) r_{u}) \times r_{v} + r_{u} \times ((D_{x} v) r_{v})$ ; and $\partial_{t} [B (r (p, t), t)] = \partial_{t} B (r (p, t), t) + (D_{x} B) r_{t}$ by the chain rule. Hence, multiplying out and evaluating the spatial derivatives at $r (p, t)$ , $\partial_{t} h = ⟨ \partial_{t} B + (D_{x} B) \tilde{v}, w ⟩ + ⟨ B, \partial_{t} w ⟩ = ⟨ \partial_{t} B + (D_{x} B) v + (div v) B - (D_{x} v) B, w ⟩$ , the last equality by step 1.2 with $A : = D_{x} v$ , $B_{0} : = B$ , $(u_{1}, u_{2}) : = (r_{u}, r_{v})$ .

3.1step 2.2F6algebra

Conversion by the cross-product identity. By [F6] with $F : = v$ and $G : = B$ , $curl (v \times B) = (div B) v - (div v) B + (D v) B - (D B) v$ , where $(D v) B = (D_{x} v) B$ and $(D B) v = (D_{x} B) v$ ; substituting into step 2.2 gives $\partial_{t} h = ⟨ \partial_{t} B + v div B - curl (v \times B), w ⟩$ .

4.1step 2.1step 3.1F7F8algebra∎

Parametric Stokes and conclusion. Fix a time $t$ of the nondegenerate $I$ . The spatial slices of $v, B$ are $C^{1}$ on the open slice of $W \cap V$ containing $r (D, t)$ ; hence $G = (v \times B) (\cdot, t)$ is $C^{1}$ there. Restrict the $C^{2}$ extension of $φ = r (\cdot, t)$ to an open neighbourhood $O \supseteq D$ mapping into that slice. The first supplier in [F7] makes $P^{*}, Q^{*}$ $C^{1}$ on $O$ with the displayed curl integrand. Green then gives its integral over $D$ as the positive boundary integral, and the pullback supplier identifies each arc integral with $\int_{φ \circ σ_{ℓ}} G \cdot d r$ . Thus $\int_{D} ⟨ curl (v \times B) (r, t), w ⟩ d p = \oint_{\partial S (t)} (v \times B) \cdot d r$ without requiring a published regular patch. Substituting into steps 2.1 and 3.1 gives the always-defined parameter identity $Φ^{'} (t) = \int_{D} ⟨ \partial_{t} B (r, t) + v (r, t) div B (r, t), w ⟩ d p - \oint_{\partial S (t)} (v \times B) \cdot d r$ . At a rank-two slice its first two integrals are exactly the surface shorthand defined in the Statement (and, for an ordinary regular patch, [F8]); if $div B = 0$ the middle term vanishes. This proves the stated identity with the minus sign, including relative endpoint derivatives whenever the normal shorthand applies.

Remarks

Sign bookkeeping. The boundary term appears with a minus sign because Stokes converts the flux of the curl into the circulation in the induced orientation; the additional term $(v \cdot n) div B$ vanishes whenever the field is divergence-free. The example on the companion page checks both signs by direct computation.
Hypotheses used. The $C^{2}$ regularity of $r$ is used for the equality of mixed partials, the single-valued $C^{1}$ velocity makes $\tilde{v}$ well defined, and the nonvanishing of $r_{u} \times r_{v}$ on the interior identifies the orientation; the velocity is given on one open neighbourhood of the whole swept graph, and boundary chains may have the corners permitted by the finite elementary decomposition.

Transport of flux through a moving patch

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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