By Charles Bradfield Morrey Jr.

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The left-hand side, ∂T /∂t, is quite simple. It is the rate of change of the temperature T (x, y, z, t) when the spatial coordinates x, y, and z are kept ﬁxed and t varies. In other words, it tells us how fast the point (x, y, z) is heating up or cooling down at time t. What would we expect this to depend on? Well, heat takes time to travel through a medium, so although the temperature at some distant point (x , y , z ) will eventually aﬀect the temperature at (x, y, z), the way the temperature is changing right now (that is, at time t) will be aﬀected only by the temperatures of points very close to (x, y, z): if points in the immediate neighborhood of (x, y, z) are hotter, on average, than (x, y, z) itself, then we expect the temperature at (x, y, z) to be increasing, and if they are colder then we expect it to be decreasing.

Here, we would like to say that the numbers approach 0, even though it is not true that each one is closer than the one before. Nevertheless, it is true that eventually the sequence gets as close as you like to 0 and remains at least that close. This last phrase serves as a deﬁnition of the mathematical notion of a limit : the limit of the sequence of numbers a1 , a2 , a3 , . . is l if eventually the sequence gets as close as you like to l and remains that close. 2]. Suppose δ is a positive number (which one usually imagines as small).

More generally, however accurate we wish our estimate of π 2 to be, we can achieve this accuracy if we are prepared to make x a suﬃciently good approximation to π . In mathematical parlance, the function f (x) = x 2 is continuous at π . Let us try to say this more symbolically. The statement “x 2 = π 2 to within an accuracy of ” means that |x 2 −π 2 | < . To capture the phrase “however accurate,” we need this to be true for every positive , so we should start by saying ∀ > 0. ” The thought behind them is that there is some δ > 0 for which the approximation is guaranteed to be accurate to within as long as x is within δ of π .