TITLE: Newton's law of cooling for the heat equation boundary condition
QUESTION [9 upvotes]: Newton's law of cooling says the temperature of an object satisfies
$$
\frac{dT}{dt} = -k(T(t) - T_0),\tag{1}
$$
where $T_0$ is the surrounding temperature. See these HTML notes for example.
Now if $u(x,t)$ denotes the temperature of a laterally insulated bar at a point $x$ and time $t$, then heat equation says
$$
\frac{\partial u}{\partial t} = \alpha\, \frac{\partial^2 u}{\partial x^2}
$$
If the left end $x = 0$ is exposed to an environment at temperature $T_0$, all mathematical physics books say that Newton's law of cooling is
$$
-c \frac{\partial u}{\partial x}(0,t) = -k( u(0,t) - T_0), \tag{2}
$$
where $c$ is a constant. See Problem 5 in this Google book preview of Boundary Value Problems: and Partial Differential Equations by David Powers. See also page 131 of the same book. (Three pages before the problem 5, where Newton's law of cooling first appears.)
However, according to Newton's law of cooling (1), we get
$$
\frac{\partial u}{\partial t}(0,t) = -k( u(0,t) - T_0),
$$
By the heat equation, $\frac{\partial u}{\partial t} = \alpha\, \frac{\partial^2 u}{\partial x^2}$, we conclude that
$$
\alpha \frac{\partial^2 u}{\partial x^2}(0,t) = -k( u(0,t) - T_0).\tag{3}
$$
This boundary condition is not the same as (2). Why is (3) not correct?

REPLY [1 votes]: Newton's law of cooling actually comes from the more general equation for heat $Q$ transferred between a system (temeperature $T$) and it's surroundings(temperature $T_0$):
$$\frac{dQ}{dt} = -hA(T-T_0)$$
where $A$ is the area through which heat transfer occurs (see, for example, here). For an ordinary macroscopic object, where $dQ = mc\ dT$, we get the conventional Newton's law of cooling in terms of temperature:
$$mc\frac{dT}{dt} = -hA(T-T_0)$$
For the case of the conducting bar however, from Fourier's law:
$$\frac{1}{A}\frac{dQ}{dt} = -k\frac{\partial u}{\partial x}$$
The boundary condition is therefore:
$$-k\frac{\partial u(0, t)}{\partial x} = -h(u(0, t) - T_0)$$


TITLE: Construction of appropriate Morse functions
QUESTION [4 upvotes]: I am interested in the properties of connectedness of level sets of Morse functions. Let $M$ a compact smooth $n$-manifold, and $1\leq k<n$. Is it possible to construct $k$ Morse functions $f_1,\cdots,f_k:M\rightarrow [0,1]$ on $M$ such that for $\mathscr{L}^k$ almost every $t\in [0,1]^k$,
$$
\bigcap_{j=1}^k f_j^{-1}\{t_j\}
$$
is a connected $(n-k)$ submanifold? (of course $\mathrm{Im} f_j=[0,1]$) The case $k=n-1$ is of primary interest for me, and even $n=2$ would be interesting. For example, il we take a $2$-sphere, if $x_3$ if the third coordinate in $\mathbb{R}^3$ then except in $-1$ and $1$, we have a connected curve, but if we take for example $f(x)=(x_3-\frac{1}{2})^2$, then $f^{-1}\{t\}$ is the union of two circles, the equator if $t=\frac{1}{2}$ or the empty set, so we cannot choose arbitrary Morse functions in this construction, if it is possible.

REPLY [2 votes]: For $n>1$ and $k=1$ this is possible (contrary to a comment above). Choose a Morse function that has a single minimum and a single maximum (I assume your $M$ is connected!).  If $n=2$, assume also that you pass all of the index $1$ critical points simultaneously (ie each the index $1$ critical level is one point). Then the level set just above the index $1$ critical level is still connected, even in dimension $2$ (paradoxically, the hardest case for this argument). So for $n=2$ we are done.
It's probably easier to visualize this case ($k=1$) in terms of handles; I'm just saying to add the $1$-handles simultaneously in dimension $2$.  Recall that passing an index $d$ critical point means that you do a surgery on a $d-1$ sphere, and this doesn't disconnect the level set unless $d=n-1$. So after passing the critical levels of index $< n-1$, you have a connected level set. On the other hand, if there's only one $n$-handle (one maximum) then for $n>2$ the level set after any $n-1$ handle is still connected.
You might be able to do the case $n=3$, $k=2$ by similar naive arguments, ie choose a Morse function on your 3-manifold where there is only one index $1$ (resp. $2$) critical level, and then look for a family of Morse functions on the family of surfaces sweeping out your 3-manifold, each with connected level sets.