Q&A: Basis vectors are exactly the same as partial derivative operator? 为什么向量等价于微分算子?
1 Question
In differential geometry, we usually see a vector \(v\) is written as: \[ v = v^i \frac{\partial }{\partial x^i} \bigg\rvert_p. \]
Why does a vector naturally relates to partial derivatives?
2 Solution – From Derivative to Derivation
2.1 Directional derivative
We know from multivariable calculus that in high dimensions, we could not say the “derivative”, but the directional derivative of a function1. The directional derivative is a measure of how quickly the function value vary when we step a tiny nudge along a vector \(v\). Imagine we are at \(p\) in \(\mathbb{R}^3\) and temperature is different everywhere. We are curiously about how this temperature field \(f\) changes in different directions. we move a tiny proportion2 along \(v\) (say \(\epsilon = 0.01 \%\)) and we feel the temperature changes by \(\Delta f = f(p+\epsilon v)-f(p)\). So we define the directional derivative of \(f\) along \(v\) is \[ D_{v} f |_p := \lim_{\epsilon \to 0} \frac{\Delta f}{\epsilon}. \]
1 “Scalar field” in fancier term. A scalar field in \(\mathbb{R}^n\) is a map from \(\mathbb{R}^n\) to \(\mathbb{R}\).
2 This is important! We are NOT moving a tiny bit but a tiny proportion, which means the length of \(v\) matters. Because if we move \(0.01 \%\) on \(v\) and \(2 v\), \(f\) will vary \(\Delta f\) and \(2\Delta f\) and therefore the directional derivative of \(f\) along \(2v\) would be doubled! In some books, you will see we force \(v\) to be unit length, so we will not have this problem. But for me it’s unnecessary.
3 We use upper indices to represent coordinate components and lower indices to represent basis vectors, so Equation 1 in usually notation is just \[ D_v f = \langle \frac{\partial f}{\partial x} \hat{\imath} + \frac{\partial f}{\partial y} \hat{\jmath} +\frac{\partial f}{\partial z} \hat{k}, v_1 \hat{\imath} + v_2 \hat{\jmath} + v_3 \hat{k} \rangle. \]
It turns out that there is an explicit formula for directional derivatives: \[ D_{v} f = \langle\nabla f, v\rangle, \] i.e., the inner product between the gradient of \(f\) and \(v\). The direction of the \(\nabla f\) is the steepest ascend of \(f\) at \(p\). In \(\mathbb{R}^3\), this can be written as3 \[ \begin{aligned} D_v f &= \langle \frac{\partial f}{\partial x^1} e_1 + \frac{\partial f}{\partial x^2} e_2 +\frac{\partial f}{\partial x^3} e_3, v^1 e_1 + v^2 e_2 + v^3 e_3 \rangle \\ &= v^1 \frac{\partial f}{\partial x^1} + v^2 \frac{\partial f}{\partial x^2} + v^3 \frac{\partial f}{\partial x^3} \\ &= \sum_i v^i \frac{\partial f}{\partial x^i} \\ &=: v^i \frac{\partial f}{\partial x^i}. \end{aligned} \tag{1}\]
The last step in Equation 1 where we drop the summation notation is a convention called Einstein notation.
We could view \(D_v f\) as \(v\) acts on \(f\). Some textbook uses \(v[f]\) to represent this action, i.e., \[ v[f] := D_v f. \]
2.2 Derivation
We know a normal derivative satisfy so-called chain rule: \[ \frac{\mathrm{d}}{\mathrm{d}x}(fg) = \frac{\mathrm{d}f}{\mathrm{d}x} g + f \frac{\mathrm{d}g}{\mathrm{d}x}. \]
We extract this property and define abstractly the derivation operator on an algebra as follows:
It’s obvious that every \(v\) induces such a derivation on the algebra \(C^\infty_p\) by a map \(\phi: v \mapsto D_v\). The question is: Does every derivation necessarily induced by a vector?
In other words, every possible derivations on the algebra \(C^\infty(\mathbb{R}^n)\) is some directional derivative along \(v \in T_p (\mathbb{R}^n)\). Under this isomorphism, the basis vectors \(e_i\) is mapped to the partial derivative operator \(\frac{\partial }{\partial x^i}\)!
In a general manifold \(M\), we actually use derivations to define tangent vectors on a manifold4. Because the concept of derivations are just functions that satisfy certain property, which is easy to define. While vectors seem exclusively belongs to Euclidean space. So :
4 Tu’s book is a very good book of differential geometry for beginners, check it out!
This is common in mathematics. We call this “stereotyping”, ah sorry, “abstraction”. We find two similar concepts (e.g. vectors and directional derivatives) on some object (euclidean space). But one of them (directional derivative) can be easily generalized to another objects (“manifold”). So then Mathematicians use some of its properties back to define itself axiomatically and called it the same name just to confuse people (“tangent vectors”)5. Or invent another name (e.g. topological space) just to be intimidating. Anyway, you will feel comfortable once you get used to them.
5 Other examples include topological spaces, groups, \(\sigma\)-algebra, “measurable spaces”, etc. These are just abstraction of open sets, closed stuff, events, volumes, etc.