I have $T(x,y)$ (a function of two variables $x$ and $y$) and it is the temperature distribution on a metal plate.
What does temperature distribution mean? what does its equation say? if I have the equation of the temperature distribution of the plate, can I find the equation of the plate itself? or can I say anything about the plate?
Context of the question: $T:[0,1]\times[0,1]\to\mathbb{R}$ given by $T(x,y)=x^2+y^3$. A robot traverses a path $c:[0,1]\to\mathbb{R}^2$ given by $c(t)=\left(t,\sin\left(\frac{t\pi}{2}\right)\right)$. I need to explain why the robot remains on the plate but I do not have the equation of the plate and I am puzzled.
Best Answer
Background: The equation describing the temperature distribution in a piece of material is usually called heat equation. It is a partial differential equation (PDE) and has two main forms:
(A) time-dependent heat equation: $$ \frac{\partial u}{\partial t} = \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right). $$
(B) steady-state heat equation $$ \alpha\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0. $$ The coefficient $\alpha$ characterizes the material and is called thermal diffusivity; $u$ is the temperature; $x,y,z$ are spatial coordinates, and $t$ is time. The thermal diffusivity $\alpha$ may be constant or it may depend on coordinates and/or time. An additional term representing internal heat generation may also appear in the equation.
To solve the heat equation and find the temperature distribution, we need boundary conditions on the surface of the material as well as, in case (A), the initial condition (i.e. the temperature distribution at the time $t=0$).
The $T(x,y)$ in the problem statement gives temperature as a function of spatial coordinates $x,y$; therefore $u=T(x,y)$ must be a solution of the heat equation (for some $\alpha$ and some particular form of internal heat generation). So in this problem you do not need to solve the heat equation because you are already given a particular solution (spatial distribution of temperature): $T(x,y)=x^2+y^3$ in the square plate $x\in[0,1]$, $y\in[0,1]$.
In this particular problem about a robot's motion: The robot's path is already given as a parametric curve $c(t)=(x(t),y(t))=(t,\sin{t\pi\over2})$. The parametric equation $(x,y)=(t,\sin{t\pi\over2})$ maps the parameter $t\in[0,1]$ to coordinates $(x,y)$. The given parametric equation of the robot's path guarantees that both $x$ and $y$ are within $[0,1]$ because $$ \qquad\qquad x = t, \ \ t\in[0,1] \quad\Rightarrow\quad x\in[0,1], $$ $$ \ y = \sin(t\pi/2), \ \ t\in[0,1] \quad\Rightarrow\quad y\in[0,1]. $$
So in this (part of) problem, there is apparently no need to solve (or even know) the general form of the heat equation.