Why 2nd Derivative
Derivation
Math
Explanation
E is the total heat energy in the region R
Here u is the heat in the region - making sense that the integration would give the total heat energy.
Conservation of energy says that this is true:
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Where Q is the heat content which is equal to the internal energy of a system plus the product of its volume and pressure (also known as the local heat flux density); and nis the normal vector. This is saying that the change in heat energy in time must be the same as the heat being put in or out of the system.
Fourier's Law says that this is true:
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Where k is the material’s conductivity and grad(U) is the temperature gradient
The temperature gradient is saying in which direction and how rapidly the temperature is changing
Divergence demonstrates the fluid flow and the direction of the flow
Seeing both of these components together as operators on U is called the Laplace operator. This is used for diffusion purposes, which is behind the idea of the heat equation having all the curves and bumps flatten out over time.
Therefore this makes sense that there is a second derivative since the temperature will be dependent on the curve of the temperature.