Paper 3, Section II, A

Vector Calculus | Part IA, 2007

(i) Define what is meant by a conservative vector field. Given a vector field A=(A1(x,y),A2(x,y))\mathbf{A}=\left(A_{1}(x, y), A_{2}(x, y)\right) and a function ψ(x,y)\psi(x, y) defined in R2\mathbb{R}^{2}, show that, if ψA\psi \mathbf{A} is a conservative vector field, then

ψ(A1yA2x)=A2ψxA1ψy\psi\left(\frac{\partial A_{1}}{\partial y}-\frac{\partial A_{2}}{\partial x}\right)=A_{2} \frac{\partial \psi}{\partial x}-A_{1} \frac{\partial \psi}{\partial y}

(ii) Given two functions P(x,y)P(x, y) and Q(x,y)Q(x, y) defined in R2\mathbb{R}^{2}, prove Green's theorem,

C(Pdx+Qdy)=R(QxPy)dxdy\oint_{C}(P d x+Q d y)=\iint_{R}\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right) d x d y

where CC is a simple closed curve bounding a region RR in R2\mathbb{R}^{2}.

Through an appropriate choice for PP and QQ, find an expression for the area of the region RR, and apply this to evaluate the area of the ellipse bounded by the curve

x=acosθ,y=bsinθ,0θ2πx=a \cos \theta, \quad y=b \sin \theta, \quad 0 \leqslant \theta \leqslant 2 \pi

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