Naturalness of the cartesian coordinate system

Question

The polar and spherical coordinate systems are intuitive when it comes to locating a point on a plane or in a space but they lack the naturalness of the cartesian coordinate system when it comes to calculus and especially integration. The following integral

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

corresponds to the area under the curve of $f$ in the cartesian coordinate system and can be computed using antiderivatives.

However, the area under the curve of $f$ in the polar coordinate system is given by

$$\int_{a}^{b} \frac{f(\theta)^2}{2} \, d\theta$$

and we lose the nice relationship between area and antiderivatives. Is this why the cartesian coordinate system is considered standard?

Answer 1

I wouldn't say there is a clearcut superiority of one system over other systems.

Each time you have a system of two mutually orthogonal curves, like circles and radials when you use polar representation, or other examples such as a doubly parabolic one, or this one with hyperbolic curves one must separate different uses:

• For spotting a point, polar representation for example can be more natural than cartesian system: "I see this point at such azimuth (sometimes said "at 2 o'clock"...) at such distance".

• For computations, as said for example integration, the formula you give looks more complicated, but have you thought that converting the $r=f(\theta)$ into its cartesian counterpart can be rather complicated (some curves have a much more simple expression in polar coordinates). It is a question of balance...

• For understanding phenomenas: seeing them in their natural "setting" (which can be a system of curves like for electrical fields here.

However, in the polar coordinate system, the area under the curve of a function $f(\theta)$ in the closed interval $[a,b]$ does not correspond to the definite integral $\int_a^b f(\theta) \; d\theta$.

Thus, the area under the curve of $f(\theta)$ cannot be evaluated using an antiderivative $F(\theta)$.

Answer 2

This is a somewhat philosophical question. Much of the development of the calculus used the Cartesian coordinate system. It pre-dated others. Questions of naturalness of a coordinate system and a "standard" system may be dependent on this history, what someone was taught first, and the context in which the system is used. For example, in 2D computer graphics, homogeneous coordinates introduced in 1827 are used frequently because they enable translations to be expressed using matrices.