Let be a rectangle in the plane

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 R=[0,a]×[0,b]R = [0,a]\times[0,b] be a rectangle in the plane, and let F(x,y)=(P(x,y),Q(x,y))\mathbf F(x,y) = (P(x,y), Q(x,y)) be a vector field with continuous partial derivatives on the closed rectangle (including its boundary). ((a)) Prove, by integrating each term of ∇⋅F=∂P/∂x+∂Q/∂y\nabla\cdot\mathbf F = \partial P/\partial x+\partial Q/\partial y in its own variable and identifying the resulting edge integrals, that ∬R(∂P∂x+∂Q∂y)dA=∮∂RF⋅n^ ds,\iint_R \left(\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\right)\mathrm dA = \oint_{\partial R}\mathbf F\cdot\hat{\mathbf n}\,\mathrm ds, where n^\hat{\mathbf n} is the outward normal to ∂R\partial R. Name the hypothesis on F\mathbf F used at each step. ((b)) A thin rectangular plate occupying RR is thin enough that its temperature u(x,y)u(x,y) does not vary through its thickness, and uu is twice continuously differentiable on the closed rectangle, boundary included. It satisfies the steady two-dimensional Poisson equation ∇2u=−f(x,y)/k\nabla^2 u = -f(x,y)/k inside RR, where ff is the internal heat-source density and k>0k>0 is the (uniform) conductivity, with uu prescribed on the whole boundary ∂R\partial R (a Dirichlet condition). Using the result of part (a) applied to the field w∇ww\nabla w, where ww is the difference of two solutions with the same ff and the same boundary data, prove that uu is unique.

Browse the problem library