Describe the laplace function to someone who has not studied maths?

Question

I would like to understand this but from what I have found online many of the explanations use a lot of mathematical terminology I am not too familiar with. For someone who has just a a basic to intermediate understanding of math, What is the Laplace function and what does it do?

Answer 1

I'm going to assume you mean the Laplace operator, often written $$ \Delta = \nabla^2 = -\bigg(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\bigg) $$

The Laplace operator measures how much a quantity deviates from equilibrium.

In many physical models, we have some quantity that varies over space. We would like to understand how that quantity changes over time, based on how it varies over space. When a quantity tends to move toward equilibrium, we can use the Laplace operator to measure how far it is from equilibrium and use that measurement to describe how it changes over time.

Let me give an example. When you put your spoon in a cup of hot tea and leave it for a little bit, it becomes hot to the touch. The spoon starts at room temperature, but part of the spoon is immersed in the hot tea. We can describe temperature as a function of location on the spoon - different locations on the spoon have different temperatures.

Temperature seeks equilibrium. The further out of equilibrium temperature is, the faster it moves toward that equilibrium. So at each point of the spoon, the temperature moves toward its equilibrium. This leads to the spoon warming up, to the temperature of the hot tea.

We can write these dynamics precisely by supposing that $T$, temperature, is a function of position $(x,y,z)$ and of time $t$. The evolution of $T$ over time will be governed by the equation $$ \frac{\partial u}{\partial t} = c\Delta u $$ At each location $(x,y,z)$, the further $u$ is away from equilibrium, the faster it moves toward equilibrium.