The most fundamental theory of the universe known to man start with a box and a string. Yeah yeah i know it sounds pretty crazy but let me explain.
Imagine that you have a box on a string and that you pull on the box and then release it. Then you will find that the box start to oscillate back and forth.
The mathematics of this phenomena is very easy to understand. Using Taylor series we can find that oscillation forces will behave like this if the region is small enough no matter what the real force is.
This phenomenon is used to explain parts of particle physics and quantum mechanics. These theories are used to explain anything from how light can travel in space, to why atoms don't break. So if you wanna understand why the universe exist you can bet yourself that you wanna understand Taylor series.
Taylor series have many more applications in areas such as finance, biology, medicine and chemistry.
Back in single variable calc, we used Taylor's theorem to approximate a function around a given point, using a polynomial. Well, surprise surprise, Taylor's theorem for multiple variables does just the same thing.
Taylor expansions approximate functions as polynomials, in one or many dimensions alike
Pick a point. Compute a bunch of derivatives. Plug it into the Taylor formula. And voilà, that's your polynomial.
The idea here is that the approximation and the original function have the same partial derivatives.
To see an example, let's approximate the function
using a Taylor expansion of order two around the point .
The formula for around a point is
We proceed by calculating all the required derivatives
Finally, our Taylor approximation is
Setting and , we can take a look on how the approximation looks like below
Sometimes you run into wacko functions, like . I don't know about you, but I don't have an intuitive feel for how this thing behaves. I mean, it contains two separate terms. It's the kind of function which might give you nightmares.
But here's the key: we can easily compute the partial derivatives, right? The partial derivatives help us wrap our heads around , at least if we zoom in on a small area around a given point. Actually, the tangent plane is a way of approximating a function. Think about the differentials! As you move further from the point, the change increases. The tangent plane isn't an accurate approximation though, since it only consists of first-order terms.
But we could also draw on information about the second order partial derivatives, improving our approximation. Then you'd get something like
If we're close to the point, where , the higher order terms will be smaller. They won't matter as much.
We can, as a matter of fact, apply the same reasoning to a function of more than variables, getting
Find the second order Taylor approximation of about the point
the function has the following partial derivatives
note that all the terms with a will evaluate to at the point , plugging this into the formula yields
which is the function we can see in the image above.