For a point in -dimensional space and a fixed length

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

For a point

r=0

n

-dimensional space and a fixed length

a>0

, decide the convergence of

\int_{r\le a} r^{-p}\,\mathrm{d}^n x

(near the origin) and

\int_{r\ge a} r^{-p}\,\mathrm{d}^n x

(out to infinity) for each of the following cases, all purely mathematical and dimensionless. ((a))

n=1

p=1.5

: both integrals. ((b))

n=2

p=1.5

: both integrals. ((c))

n=4

p=3

: both integrals. ((d))

n=4

p=5

: the integral out to infinity only.

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