Which is more elegant in physical analysis, Calculus or Algebra?

[Dbjergaard]

I have been studying physics for two years. (High school physics supplemented by the Feynman lectures, as well as an algebra based freshman course) Lately, I've been looking of various derivations of things (escape velocity, kinematic equations etc.) and wondering which is more elegant in a physical proof of a concept or equation. Is it calculus or algebra? Calculus, in it self is elegant, but in algebra is more simplistic, and is accessible to a wider audience. I'm slightly partial towards calculus as being more elegant, but I could be swayed either way. What are your thoughts?

[Claude Bile]

Elegance is a tough thing to quantify. If we define "elegance" as the number of steps used to reach a solution or proof, then I would venture that calculus could probably be regarded as a more elegant method in most circumstances (but not all). If however, you define elegance as a measure of how easy the method is to understand, then algebra might come up trumps if the target audience do not have a strong understanding of calculus.

Elegance though in my opinion is a subjective, almost aesthetic virtue in mathematics.

Claude.

[Gokul43201]

Sometimes the algebraic approach is much more elegant than a calculus approach. Case in point - selection rules for transitions.

[robphy]

Sometimes the geometrical solution is the most elegant... it really depends.

This was a problem from a mathematical problem solving class in college:

Consider two vertical flagpoles, with heights h_A and h_B, separated by a distance d on level ground.
Locate a point C on the ground such that the total length of an ideal rope from the top of A to C on the ground, back up to the top of B is as short as possible.

The prof led us through three solutions... calculus, algebraic, and geometrical.
The calculus and algebraic solutions took a few boards. Then he said, "Geometric solution: 2 seconds!"

[Gokul43201]

And then there's intuition - typical time < 1 sec. No need to flip a flagpole...just go with similar triangles.

[Chronos]

Math, like phyics, is very inelegant in classical terms. Algebra is deterministic, the universe more resembles a poorly defined calculus problem.