Understanding Coordinate transformation under tensor calculus

Question

I am reading the book "Tensor Calculus" by Schaums Outlines and I came across this paragraph.

Suppose that in some region of $\mathbb{R^n}$ two coordinate systems are defined and these two systems are connected by equations of the form $$f: \bar x^i =\bar x^i (x^1, x^2,...,x^n)$$ where $(1 \leq i \leq n)$

where for each i, the function, or scaler field, $x^i (x^1 , x^2 ,...,x^n)$ maps the given region in $\mathbb{R^n}$ to the reals and has continuous second partial derivatives at every point in the region.

The transformation $f$ if bijective is called a coordinate transformation.

I am struggling with the basic intuition of the above paragraph.

What does it mean to say two coordinate systems are defined?

What is the equation $f: \bar x^i =\bar x^i (x^1, x^2,...,x^n)$ saying? And how does $f: \bar x^i =\bar x^i (x^1, x^2,...,x^n)$ connect these two systems?

What does $x^i (x^1 , x^2 ,...,x^n)$ represent?

Answer 1

A good canonical example to keep in mind is the plane $\Bbb{R}^2$, with the two coordinate systems given by

• Cartesian (rectangular) coordinates
• Polar coordinates

So the same point in $\Bbb{R}^2$ can be written as $(x,y)$ or as $(r,\theta)$. In the notation you mentioned, think of $(\bar{x}^1, \bar{x}^2) = (x,y)$ as one coordinate system, and $(x^1,x^2) = (r,\theta)$ is the second coordinate system. Then the map $f$ just says how to convert from one to the other. In this example of polar coordinates, we'd have

$$ \bar{x}^1(x^1,x^2) = x(r,\theta) = r \cos(\theta) $$ $$ \bar{x}^2(x^1,x^2) = y(r,\theta) = r \sin(\theta) $$