Hi Max,

Suppose you have a piece of some material, maybe a concrete block, a piece of wood or perhaps a stone. Suppose this material is in 3-dimensional space and I can determine the co-ordinates of any point in this material. Let $u$ be the temperature of this material in degrees $C$ at the point $(x,y,z)$ at time $t$ seconds. Hence $u$ is a function of 4 variables so I can write $u$ as $u(x,y,z,t).$

In the heat equation

\[\frac{\partial u}{\partial t} = a \bigtriangledown^2 u\]

the left side is the partial derivative of $u$ with respect to $t.$ This is the rate at which the temperature is changing over time at each point and is in the units of degrees Celsius per second.

On the right side $a$ is a property of the material called the thermal diffusivity, and

\[\bigtriangledown^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}.\]

The expression

\[\frac{\partial u}{\partial x}\]

is the partial derivative of $u$ with respect to $x$ and measures the rate at which the temperature is changing as you move in the $x$ direction.

The expression

\[\frac{\partial^2 u}{\partial x^2}\]

is the second partial derivative of $u$ with respect to $x$ and measures the rate at which the temperature change is changing as you move in the $x$ direction. It's a kind of acceleration or curvature of the temperature function $u$ as you move in the $x$ direction.

Taken together

\[\bigtriangledown^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\]

measures the "curvature" of $u.$

Usually the people who work with the heat equation are trying to solve for $u$ under various conditions.

I hope this helps,
Penny