Lagmental Vicfred

Tangent Vectors Are Derivations at a Point by Vicfred

tags: mathematics, smooth manifolds, tangent spaces, differential forms

A tangent vector can be defined intrinsically as a Leibniz derivation on germs of smooth functions. I want the notation, the mechanism, and the failure mode visible at the same time.

Notation

A smooth \(n\)-manifold is locally modeled on \(\mathbf R^n\) with smooth transition maps. A smooth map \(F:M\to N\) differentiates to linear maps between tangent spaces.

$$ v:C^\infty_p(M)\to\mathbf R $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ v(fg)=f(p)v(g)+g(p)v(f) $$

Stress the formula

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ v=\sum_{i=1}^{n}v^i\left.\frac{\partial}{\partial x^i}\right|_p,\qquad v(f)=\sum_i v^i\frac{\partial f}{\partial x^i}(p) $$

Interpretation

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad v:C^\infty_p(M)\to\mathbf R,\\[5pt] \mathsf{C}\;&:\quad v(fg)=f(p)v(g)+g(p)v(f). \end{aligned} $$

Limit of the argument

Coordinates are computational tools, not intrinsic data. Tensorial formulas must transform correctly on chart overlaps.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] v(fg)=f(p)v(g)+g(p)v(f) \end{gathered}} $$

The result is compact enough to reuse without pretending that the caveat has disappeared. The worked line remains the quickest consistency check.

This article was posted on Sun 29 April 2012. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.