Choice of norm in multivariable calculus theory

Question

In $\mathbb{R}^n$ the euclidean norm of $x \in \mathbb{R}^n$ is the number $||x|| \in \mathbb{R} $ such that $||x||=\sqrt{(x_1^2+x_2^2+...+x_n^2)}$.

It induces the euclidean distance: if $x,y$ are points in $\mathbb{R}^n$, the euclidean distance between them is the number $d(x,y)=||x-y||=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+...+(x_n-y_n)^2}$.

This distance is the one that has been used in all the multivariable calculus theory that I have studied this year. There are lots of norms that induce distances, so why has the euclidean norm been the chosen one? Do other norms induce a "worse" distance in some sense for multivariable calculus theory construction?

Answer 1

There is no a priori reason to use the Euclidean norm on $\mathbb{R}^n$ when doing multivariable calculus. From an instructional point of view, there are many reasons to take this approach for an introductory course in multivariable calculus:

• Most of the students in such courses are engineers, who will generally be building structures on the surface of the Earth. Sure, relativity and the curvature of space time matter, but using a local model of $\mathbb{R}^3$ with the Euclidean norm is good enough for most engineering tolerances. Building bridges that don't fall down and radios that work is hard enough without additionally worrying about other choices of norm.
• The point of an introductory course in multivariable calculus is to become familiar with integration and differentiation in several variables and generally capstones with Green's Theorem, Stokes' Theorem, and the Divergence Theorem (with the associated gradient, curl, and divergence for vector fields in $\mathbb{R}^3$). These concepts are usually hard enough, without needing to worry about the choice of norm.
• The Euclidean norm corresponds to the standard dot product, which in turn dictates things like projections. Choosing a different norm requires some (strong) familiarity with (multi-)linear algebra.
• In principle, we could teach multivariable calculus with do Carmo's Riemannian Geometry and Halmos's Measure Theory rather than with Rogawski or Stewart. In practice, almost everyone would fail the course if this was their first introduction to multivariable calculus. Similarly, physics is easier to learn by starting with Newtonian mechanics (which generally is $\mathbb{R}^n$ and the Euclidean norm) and then later learning about relativity and quantum mechanics.

As for what multivariable calculus looks like when we start tinkering under the hood:

• First off, note that all vector norms on a finite dimensional space are equivalent. However, the geometry will radically change. With respect to the Euclidean norm, the sphere $S^n$ is a nice smooth surface. What happens when you start tweaking the norm? Note that the infinity norm is not induced by a dot product, which makes things like projections rather problematic.
• In Riemannian Geometry, we have a continuously varying choice of how to take inner products at each point, and then define distances in terms of length minimizing geodesics. In effect, we might have the Euclidean norm in one region and then pass through various $L^p$ norms (or even more exotic options) as we move along a path.
• If you want to study vector fields in $\mathbb{R}^{n}$ for $n > 3$, the familiar curl and divergence go away, as well as the cross product. We need to start studying tensor calculus and the exterior derivative. This simplifies some things (e.g., generalized Stokes' Theorem instead of Green's / Stokes' / Divergence), but complicates many others (the linear algebra becomes much more difficult).