====================================%========================= \magnification=\magstep2 \baselineskip=18truept \vsize=10truein \parskip=12truept \parindent=0pt \voffset=-0.5truein \centerline{\bf MULTIPLE INTEGRALS} \bigskip \centerline{Hilary Term --- Prof J M Yeomans} \bigskip \bigskip 1. {\bf Repeated integrals} (a) For the following integrals sketch the region of integration and so write equivalent integrals with the order of integration reversed. Evaluate the integrals both ways. $$ \int_0^{\sqrt{2}} \int_{y^2}^2 y \, dx \, dy \; , \qquad \int_0^4 \int_0^{\sqrt{x}} y \sqrt{x}\, dy \, dx \; , \qquad \int_0^1 \int_{-y}^{y^2}x \, dx \, dy \; . $$ (b) Reverse the order of integration and hence evaluate: $$ \int_0^{\pi} \int_{y}^{\pi} x^{-1} \sin x \, dx \, dy \; . $$ \bigskip 2. {\bf Plane polar coordinates and planar mass distributions} (a) A mass distribution in the positive $x$ region of the $xy$-plane and in the shape of a semi-circle of radius $a$, centred on the origin, has mass per unit area $k$. Find, using plane polar coordinates, (i) its mass $M$, (ii) the coordinates $(\overline{x},\overline{y})$ of its centre of mass, (iii) its moments of inertia about the $x$ and $y$ axes. (b) Do as above for a semi-infinite sheet with mass per unit area $$ \sigma=k \exp-(x^2+y^2)/a^2 \quad \hbox{for} \quad x \geq 0 \; , \qquad \sigma =0\quad \hbox{for} \quad x < 0 \; . $$ where $a$ is a constant. Comment on the comparisons between the two sets of answers. Note that $$ \int_0^{\infty} \exp{(-\lambda u^2)}\,du = {1 \over 2} \sqrt{\pi \over \lambda} \; . $$ (c) Evaluate the following integral: $$ \int_0^{a} \int_0^{\sqrt{a^2-y^2}}(x^2+y^2)\, \arctan (y/x) \, dx \, dy \; . $$ \bigskip \vfill\eject 3. {\bf Jacobian matrix} The pair of variables $(x,y)$ are each functions of the pair of variables $(u,v)$ and {\it vice versa}. Consider the matrices $$ A =\left( \matrix{ \displaystyle{\partial x\over \partial u} & \displaystyle{\partial x \over \partial v} \cr & \cr \displaystyle{\partial y\over \partial u} & \displaystyle{\partial y\over \partial v}} \right) \qquad \hbox{and} \qquad B = \left( \matrix{ \displaystyle{\partial u\over\partial x} & \displaystyle{\partial u\over\partial y} \cr & \cr \displaystyle{\partial v \over \partial x} & \displaystyle{\partial v \over\partial y}} \right) \; . $$ (a) Show using the chain rule that the product $AB$ of these two matrices equals the unit matrix $I$. (b) Verify this property explicitly for the case in which $(x,y)$ are Cartesian coordinates and $u$ and $v$ are the polar coordinates $(r,\theta)$. (c) Assuming the result that the determinant of a matrix and the determinant of its inverse are reciprocals, deduce the relation between the Jacobians $$ {\partial(u,v)\over \partial(x,y)} = {\partial u\over\partial x} {\partial v\over\partial y} - {\partial u\over\partial y} {\partial v\over\partial x} \qquad \hbox{and} \qquad {\partial (x,y)\over\partial (u,v)} = {\partial x\over\partial u} {\partial y\over\partial v}- {\partial x\over\partial v} {\partial y\over\partial u} \; . $$ \bigskip 4. {\bf Change of variable in double integral} (a) Using the change of variable $x+y=u$, $x-y=v$ evaluate the double integral $ \int\!\!\int_R (x^2 +y^2)\, dx dy $, where $R$ is the region bounded by the straight lines $y=x$, $y=x+2$, $y=-x$ and $y=-x+2$. (b) Given that $u=xy$ and $v=y/x$, show that $ {\partial(u,v)}/{\partial(x,y)}= 2y/x$. Hence evaluate the integral $$ \int\!\!\!\int \exp(-xy) \, dxdy $$ over the region $x>0$, $y>0$, ~$xy<1$, ~$1/2