Idealise every map below as continuously differentiable and one-to-one except…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Calculus of change and accumulation

Conditions of Validity, Ch Multiple integrals: Fubini, the change-of-variables theorem and integrals that do not exist

Problem

Idealise every map below as continuously differentiable and one-to-one except possibly at the single point noted. ((a)) Evaluate

I = \displaystyle\iint_{x^2+y^2\le 1} \dfrac{\mathrm{d}A}{(x^2+y^2)^{3/4}}

by changing to plane polar coordinates. State why the vanishing of the polar Jacobian at the origin does not by itself block the change of variables, and show how the excision-and-limit argument justifies the answer. ((b)) Let

\mathsf{A} = \begin{pmatrix}2&1\\1&2\end{pmatrix}

, a positive definite matrix (dimensionless entries). Using the multidimensional Gaussian integral, evaluate

\displaystyle\int_{\mathbb{R}^2} e^{-\mathbf{x}^{\mathrm{T}}\mathsf{A}\mathbf{x}}\,\mathrm{d}^2x

and the correlation

\langle x_1 x_2\rangle

under the Gaussian weight

e^{-\mathbf{x}^{\mathrm{T}}\mathsf{A}\mathbf{x}}

(normalised so that

\langle 1\rangle=1

). ((c)) Compute the volume

V_4(R)

of a

4

-ball of radius

R = 2\ \mathrm{m}

, using

V_n(R) = \pi^{n/2}R^n/\Gamma(n/2+1)

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