Vector Calculus | Part IA, 2005

Let VV be a bounded region of R3\mathbb{R}^{3} and SS be its boundary. Let ϕ\phi be the unique solution to 2ϕ=0\nabla^{2} \phi=0 in VV, with ϕ=f(x)\phi=f(\mathbf{x}) on SS, where ff is a given function. Consider any smooth function ww also equal to f(x)f(\mathbf{x}) on SS. Show, by using Green's first theorem or otherwise, that

Vw2dVVϕ2dV\int_{V}|\nabla w|^{2} d V \geqslant \int_{V}|\nabla \phi|^{2} d V

[Hint: Set w=ϕ+δ.]w=\phi+\delta .]

Consider the partial differential equation

tw=2w\frac{\partial}{\partial t} w=\nabla^{2} w

for w(t,x)w(t, \mathbf{x}), with initial condition w(0,x)=w0(x)w(0, \mathbf{x})=w_{0}(\mathbf{x}) in VV, and boundary condition w(t,x)=w(t, \mathbf{x})= f(x)f(\mathbf{x}) on SS for all t0t \geqslant 0. Show that

tVw2dV0\frac{\partial}{\partial t} \int_{V}|\nabla w|^{2} d V \leqslant 0

with equality holding only when w(t,x)=ϕ(x)w(t, \mathbf{x})=\phi(\mathbf{x}).

Show that ()(*) remains true with the boundary condition

wt+α(x)wn=0\frac{\partial w}{\partial t}+\alpha(\mathbf{x}) \frac{\partial w}{\partial n}=0

on SS, provided α(x)0\alpha(\mathbf{x}) \geqslant 0.

3/II/10A Vector Calculus

Write down Stokes' theorem for a vector field B(x)\mathbf{B}(\mathbf{x}) on R3\mathbb{R}^{3}.

Consider the bounded surface SS defined by

z=x2+y2,14z1z=x^{2}+y^{2}, \quad \frac{1}{4} \leqslant z \leqslant 1

Sketch the surface and calculate the surface element dSd \mathbf{S}. For the vector field

B=(y3,x3,z3)\mathbf{B}=\left(-y^{3}, x^{3}, z^{3}\right)

calculate I=S(×B)dSI=\int_{S}(\nabla \times \mathbf{B}) \cdot d \mathbf{S} directly.

Show using Stokes' theorem that II may be rewritten as a line integral and verify this yields the same result.

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