Em Elementary Mathematical Prerequisites — Examples

Throughout this Example and Verification, the spatial labels are local one-based names: $x_i:=x(i-1)$ for $i=1,2,3$, $e_1=(1,0,0)$, $e_2=(0,1,0)$, $e_3=(0,0,1)$, and ${\mathrm{\partial }}_i$ denotes the canonical coordinate derivative ${\mathrm{\partial }}_{i-1}$ under this relabeling. Thus the locally named $e_i$ has canonical coordinate $1$ at $i-1$ and $0$ at the other coordinates; the tuples, inner product, cross product, divergence and curl retain the canonical meanings of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, The cross product in ${\mathbb{R}}^3$ and Divergence and curl of a $C^1$ vector field. This local notation does not change the zero-based indices of Localization: vanishing integrals or fluxes force a continuous function to vanish.

Take the unit square $D=[0,1{]}^2$ as parameter region, an open interval $I\subseteq \mathbb{R}$, a function $a\in C^2(I)$ with $a>0$ everywhere, and

$r(u,v,t):=(a(t)u,a(t)v,0),B:=e_3,v(x,t):=\frac{a^{\mathrm{\prime }}(t)}{a(t)}(x_1,x_2,0)\mathrm{.}$

Then $r$ is of class $C^2$ on the open set ${\mathbb{R}}^2\times I$, the tangent vectors are $r_u=(a,0,0)$ and $r_v=(0,a,0)$ with $r_u\times r_v=(0,0,a^2)\mathrm{\ne }0$ on $D^{\circ }\times I^{\circ }=D^{\circ }\times I$, the field $B$ is constant with $\mathrm{div}B=0$, and $v$ is $C^1$ on ${\mathbb{R}}^3\times I$ with $v(r(u,v,t),t)=(a^{\mathrm{\prime }}(t)u,a^{\mathrm{\prime }}(t)v,0)=r_t(u,v,t)$; so the data satisfy the standing hypotheses of Moving parametrized surface patches and their flux. For every $t\in I$ the parametrized flux is $\mathrm{\Phi }(t)=a(t{)}^2$, hence ${\mathrm{\Phi }}^{\mathrm{\prime }}(t)=2a(t)a^{\mathrm{\prime }}(t)$; the fixed-time surface integral of ${\mathrm{\partial }}_{t}B$ is $0$; and $\mathrm{curl}(v\times B)=-2(a^{\mathrm{\prime }}(t)\mathrm{/}a(t))e_3$, so patch Stokes gives ${\oint }_{\mathrm{\partial }S(t)}(v\times B)\cdot d\mathbf{r}=-2a(t)a^{\mathrm{\prime }}(t)$, exactly the sign in the formula of Transport of flux through a moving patch. Dropping the term $-\oint (v\times B)\cdot d\mathbf{r}$ — that is, treating the moving patch as fixed — would leave ${\mathrm{\Phi }}^{\mathrm{\prime }}=0$, which the direct derivative contradicts whenever $a^{\mathrm{\prime }}(t)\mathrm{\ne }0$. All maps are of the stated regularity and no singular support is involved; the example verifies the sign convention and the necessity of the motion term, and asserts no physical law.

• When $a$ is constant, $\mathrm{\Phi }(t)=a^2$ is constant, both terms of the identity vanish, and the contrast with the fixed-patch reading disappears; the example's discriminating case is therefore any $t$ with $a^{\mathrm{\prime }}(t)\mathrm{\ne }0$.
• The computation uses no choice principle; it is finite algebra and one application of the published Stokes theorem for a patch over the unit square.