Heat Equation Derivation and Mean Value Theorem

Question

Farlow book PDEs for Scientists and Engineers pg. 27 shows derivation for Heat Equation. It starts by stating

Net change of heat inside $[x,x+\Delta x]$ = Net flux of heat across boundaries + Total heat generated inside $[x,x+\Delta x]$

and writing the conservation equation

$$\textit{Total Heat Inside} [x,x+\Delta x]= cpA \int _{ x}^{x+\Delta x}u(s,t) ds $$

Takes derivative according to time, rewrites the equation

$$ \frac{d}{dt} \int _{ x}^{x+\Delta x} c\rho A u(s,t) ds = c\rho A \int _{ x}^{x+\Delta x} u_t(s,t) ds $$

$$ = kA [ u_x(x+\Delta x,t) - u_x(x,t)] A \int _{x}^{x+\Delta x} f(s,t) ds $$

At this point he wants to get rid of the integrals, so he uses the mean value theorem which is, for a $a < \xi < b$

$$ \int _{ a}^{b} f(x) dx = f(\xi)(b-a) $$

A $\xi$ must exist within the specified interval. He applies it to the equation, and gets

$$ c\rho A u_t(\xi_1,t)\Delta x = kA[u_x(x+\Delta x, t) - u_x(x,t)] + Af(\xi_2,t)\Delta x $$

$$ x < \xi < x+\Delta x $$

This also makes sense, there are multiple $\xi$'s for two different integrals. However below, he turns two $\xi$'s into one,

$$ u_t(\xi,t) = \frac{k}{c\rho} \bigg[ \frac{u_x(x+\Delta x,t) - u_x(x,t)} {\Delta x} \bigg] + \frac{ 1}{c\rho}f(\xi,t) $$

and while $$ \Delta x \to 0 $$

he gets

$$ u_t(x,t) = \alpha^2u_{xx}(x,t) + F(x,t) $$

In this last statement, $\xi$'s are replaced by $x$. So I have three questions: First how did the author combine the two $\xi$'s, second, while $\Delta \to 0$, how he turned them into $x$. I guess it is understandable if the $\Delta x$ becomes infinitesimally small, than whatever's inside can only be $x$? That still does not explain the two $\xi$'s though.

Third question: Net flux of heat across boundary uses $u_x(x+\Delta x,t) - u_x(x,t)$, that is derivative according to space, and takes difference between two sectional endpoints. Why $u_x$? Shouldn't we use simply $u$ here?

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Answer 1

1. In certain contexts it is usual to use the same letter for different quantities. A typical example are inequalities, in which a constant $C$ may take different values. In this case, the author is probably using the same letter $\xi$ to mean "a point between $x$ and $x+\Delta x$", with the understanding that it may take different values in different terms of the equations.
2. Since $x\le\xi\le x+\Delta x$, as $\Delta x\to0$, $\xi\to x$, independently of the real value of $\xi$.
3. Remember that $u$ represents temperature; then $u(x+\Delta x,t)-u(x,t)$ is the chenge of temperature, not the change of heat flux. Newton's law says that the heat flow is proportional to the spatial gradient of temperature, which is $u_x$; hence the expresion for the heat fluxin the interval $[\,x,x+\Delta x\,]$.