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Prove that for a steady state temperature distribution in a disk, the coldest spot is on the boundary, unless the temperature is constant throughout the disk.

How is this part suppose to help? I don't see it.

$$

\int_{-\pi}^{\pi}P(r,\theta)d\theta = 1

$$

Recall from calculus that

$$

m(b - a)\leq\int_a^bf(x)dx\leq M(b - a)

$$

where $M$ and $m$ are the maximum and minimum, respectively on the interval $[a,b]$.