Let be the L-shaped lamina formed as the union of two abutting rectangles B_1=[0,0.60

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

Let Ω⊂R2\Omega\subset\mathbb R^2 be the L-shaped lamina formed as the union of two abutting rectangles B1=[0,0.60 m]×[0,0.20 m],B2=[0,0.20 m]×[0.20 m,0.50 m],B_1=[0,0.60\,\mathrm m]\times[0,0.20\,\mathrm m],\qquad B_2=[0,0.20\,\mathrm m]\times[0.20\,\mathrm m,0.50\,\mathrm m], which share the edge segment S=[0,0.20 m]×{0.20 m}S=[0,0.20\,\mathrm m]\times\{0.20\,\mathrm m\} and otherwise do not overlap, so Ω=B1∪B2\Omega=B_1\cup B_2. ((a)) Let F\mathbf F have continuous partial derivatives on Ω\Omega (including its boundary). Apply the box divergence theorem to B1B_1 and to B2B_2 separately, add the two results, and show that the flux through the shared edge SS cancels between the two boxes, leaving ∬Ω∇⋅F dA=∮∂ΩF⋅n^ ds.\iint_\Omega \nabla\cdot\mathbf F\,\mathrm dA=\oint_{\partial\Omega}\mathbf F\cdot\hat{\mathbf n}\,\mathrm ds. Name the hypothesis that makes the cancellation exact. ((b)) The plate Ω\Omega, of uniform conductivity kk, carries a steady heat-source density f(x,y)f(x,y) and reaches a steady temperature uu, twice continuously differentiable on Ω\Omega including its boundary, satisfying ∇2u=−f/k\nabla^2u=-f/k in Ω\Omega, with uu prescribed on the whole of ∂Ω\partial\Omega. Using the result of part (a) and the energy argument, prove that uu is unique.

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