Also how such inventions are created or discovered? how can someone come with these ideas?
In: Mathematics
> how can someone come with these ideas?
Fourier’s original motivation came from the study of the [heat equation](https://en.wikipedia.org/wiki/Heat_equation), which models the flow of heat within a material.
This equation is relatively easy to solve in the simple case where the initial heat distribution is a sinusoidal function, but it is not obvious how to solve it for an arbitrary initial condition. However, the heat equation also has the property that it is *linear* – which means that the sum of two solutions of the heat equation is also a solution. Therefore, if you could represent your problem as a sum of simpler problems which you can solve, then you can solve the more complex problem by simply adding them together.
Fourier showed that *any* smooth function could be represented as a (usually infinite) sum of sinusoidal functions, allowing him to solve the heat equation for any initial heat distribution (at least in principle – any actual solution computed this way is only an approximation, because you can’t add up an infinite number of terms, but you can get as close as you need to).
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