How do a transformation 'born'?

Question

Well, there are several transformations in math. Like the laplace transformation. My question is about the utility and motivation of these transformations.

Like, when we have an equation, and we duplicate both sides, divide both sides, differentiate both sides; is the math transformations useful in this way too? Like, transforming both sides of the equation can be useful in some way? I mean if we could accept transformations as 'mathematical operations'. If not, why are them so useful?

What are the motivations to create a transformation?

Can you give me the simplest transform you know, so I can think about it?

Thanks!

Answer 1

In general, think of a "transform" as a tool for converting a problem into some new form that's easier to solve. Suppose you have some difficult problem, and you have a transform that converts it into a different problem that you already know how to solve. If you can also construct a "reverse" transform, this then gives you a three-step process for solving the difficult problem:

(1) Transform the original problem into some new form that's easier to solve

(2) Solve the easier problem

(3) Reverse-transform the answer to get an answer to the original problem

Here's a very simple example: suppose I have a line and a cylinder in 3D space, and I want to calculate their intersection. One good approach is as follows:

(1) Transform the cylinder and the line so that the cylinder centerline lies along the $z$-axis. The equation of the transformed cylinder is now very simple -- just $x^2 + y^2 = r^2$.

(2) Intersect the transformed line with the transformed cylinder (which is easy)

(3) Transform the intersection points back to the original location in 3D

Things like Laplace and Fourier transforms are much more complex, of course, but the basic principle is the same.

Answer 2

In many cases, a Laplace transform (or a Fourier transform for that matter) transforms a partial differential equation into an ordinary differential equation. It also transforms an ordinary differential equation into an algebraic equation. The thinking goes that if the we can find a function whose transform is the solution to that algebraic equation, then we have found a really easy way to solve some differential equations.

Thus, transformation hinges on finding a simple way to compute inverses, or at least place them in tables for easy lookup. In the latter case, the algebraic solutions must be put into a standard form (e.g., by partial fraction decomposition). For Fourier transforms, there is near-perfect symmetry between the transforms and the inverse transforms. For Laplace transforms, however, things are a little bit more complicated, but those inverses too are easily computed.