My question is about an excerpt from Stein and Shakarchi's Fourier Analysis, in deriving the heat equation. We have heat, $H(t)$, modelled in terms of temperature at a point $x,y$ at given time $t$, $u(x,y,t)$, for a square $S$ (see below image).
My question is about the last line of the image:
Why does the heat flow through the vertical side have a factor of $h$? Why isn't the heat flow through the vertical side just $- \kappa \cdot \partial_x u(x_0+\frac{h}{2},y_0,t)$? I suppose this is heat flow at a point. In that case, why isn't the heat flow through the vertical side $\int_0^h \kappa \partial_xu(x_0 + \frac{h}{2},y,t)dy$?
It is stated that the size of the square is small, implying that the temperature $u$ does not vary significantly over the area of the square and so does does not depend strongly on the coordinates $x, y$. This property was used to derive the second equation, an approximation, in the image you provided. And so, the integral you wrote and the last equation in the image are approximately equal:
$ \kappa \int_0^{h} \partial_x u(x_0 + h / 2, y_0, t) dy \approx \kappa h \partial_x u(x_0 + h / 2, y_0, t)$