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Mathematics for Physics

186 Smooth

A property of a function under which derivatives of all degrees exist and are continuous that is used to do calculus with well-behaved functions.

definition [d] (Smooth = Infinitely Differentiable = \(C^{\infty}\) = Class \(C^{\infty}\)) A property of a real-valued function: all partial derivatives of all orders exist.

  • \(f\) is \(C^{\infty}\) on an open set \(U \subseteq \mathbb{R}^{n}\) .

where

  • \(f\) is a real-valued function.
  • \(U\) is an open subset of \(\mathbb{R}^{n}\).
  • \(\mathbb{R}^{n}\) is \(n\)-dimensional Euclidean space.
  • \(C^{\infty}\) means infinitely differentiable.

Note:

  • the same notion applies to Euclidean maps.
  • \(C^{k}\) means continuous derivatives up to order \(k\); \(C^{\infty}\) is the intersection of all \(C^{k}\).
  • smooth and infinitely differentiable are synonymous.

definition [d] (Smooth = Infinitely Differentiable = \(C^{\infty}\)) A property in differential geometry of manifolds, maps, and functions:

  • (Manifold) transition maps \(f_{VU}\) are of class \(C^{\infty}\) .
  • (Map \(f: M \rightarrow N\)) for charts \((U,\phi)\) on \(M\) and \((V,\mu)\) on \(N\), the composite \(\mu \circ f \circ \phi^{-1}\) is \(C^{\infty}\) wherever defined .
  • (Function on \(M\)) \(f \circ \phi^{-1}\) is \(C^{\infty}\) in local coordinates .

where

  • \(M, N\) are smooth manifolds.
  • \(f_{VU}\) is the transition map between overlapping charts.
  • \((U,\phi)\) and \((V,\mu)\) are coordinate charts.
  • \(\phi^{-1}\) and \(\mu\) convert between manifold points and Euclidean coordinates.
  • \(C^{\infty}\) means infinitely differentiable.

Note:

  • a tensor field is smooth when its component functions are \(C^{\infty}\) in every chart.

186.1 Elementary Example

186.1.1 Simple

Smoothness means derivatives exist at sample points. Here a quadratic polynomial is smooth on a finite sample of the line.

\[ A = \{ -1,\ 0,\ 1 \} \]

\[ f(x) = x^{2},\quad f(-1) = 1,\quad f(0) = 0,\quad f(1) = 1 \]

\[ \dfrac{df}{dx}(0) = 0 \]

where

  • \(f\) is the smooth function.
  • \(\dfrac{df}{dx}\) is its derivative.

186.1.2 General

All higher derivatives of \(f(x) = x^{2}\) exist. On a larger sample the same rule holds.

\[ A = \{ -2,\ -1,\ 0,\ 1,\ 2 \} \]

\[ f(x) = x^{2},\quad \dfrac{d^{2}f}{dx^{2}}(x) = 2,\quad \dfrac{d^{n}f}{dx^{n}}(x) = 0\ \text{for } n \ge 3 \]

where

  • \(\dfrac{d^{2}f}{dx^{2}}(x)\) is the second derivative.
  • \(\dfrac{d^{n}f}{dx^{n}}\) is the \(n\)-th derivative.

186.2 References

  1. Frankel, T. The Geometry of Physics, 3rd ed. Cambridge University Press. — \(C^{\infty}\) manifolds; smooth transition maps; differentiable functions on \(M\).
  2. Hassani, S. Mathematical Physics, 2nd ed. Springer. — smooth maps \(f:M\to N\) via \(\mu\circ f\circ\phi^{-1}\).
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — smooth and differentiable fields in applied settings.