Paper 3, Section I, C

Vector Calculus | Part IA, 2016

State the chain rule for the derivative of a composition tf(X(t))t \mapsto f(\mathbf{X}(t)), where f:RnRf: \mathbb{R}^{n} \rightarrow \mathbb{R} and X:RRn\mathbf{X}: \mathbb{R} \rightarrow \mathbb{R}^{n} are smooth ..

Consider parametrized curves given by

x(t)=(x(t),y(t))=(acost,asint).\mathbf{x}(t)=(x(t), y(t))=(a \cos t, a \sin t) .

Calculate the tangent vector dxdt\frac{d \mathbf{x}}{d t} in terms of x(t)x(t) and y(t)y(t). Given that u(x,y)u(x, y) is a smooth function in the upper half-plane {(x,y)R2y>0}\left\{(x, y) \in \mathbb{R}^{2} \mid y>0\right\} satisfying

xuyyux=ux \frac{\partial u}{\partial y}-y \frac{\partial u}{\partial x}=u

deduce that

ddtu(x(t),y(t))=u(x(t),y(t))\frac{d}{d t} u(x(t), y(t))=u(x(t), y(t))

If u(1,1)=10u(1,1)=10, find u(1,1)u(-1,1).

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