关于巴雷特·奥尼尔《半黎曼几何及其在相对论中的应用》定义10中函数对函数求导概念的疑问

Hey there! This is such a common confusion when starting O'Neill's book—he jumps right into the coordinate-independent, function-first perspective of differential geometry, which can feel alien if you're used to standard calculus. Let me unpack this for you:

First, let's anchor ourselves to the setup O'Neill lays out in the first two pages: both $\xi$ and $u_i$ are smooth functions on the manifold, not the familiar coordinate variables (like $x, t$) we use in basic calculus. That's the key context here.

When he says we're differentiating with respect to functions, not independent variables, he's referring to a coordinate-free form of differentiation that operates on the space of functions itself. Here's a way to think about it intuitively:

• In regular calculus, you differentiate with respect to a coordinate (say $x$) to see how a function changes as you move along that coordinate axis.
• Here, we're treating each function ($u_i$ or $\xi$) like a "direction" in a bigger space of all possible functions. Differentiating with respect to $u_i$ means we're looking at how the expression in Definition 10 changes when we slightly perturb the function $u_i$ (shift its values across the manifold) while holding all other functions fixed.

The diagram you mentioned helps make this tangible: it shows how each function maps every point on the manifold to a real number. So instead of tying differentiation to a specific coordinate system's axes, we're tying it to these global functions—this is what makes the math coordinate-independent, which is critical for relativity (since spacetime doesn't have a "natural" coordinate system).

If you want to dig deeper, re-read the first chapter's section on smooth functions and tangent spaces. O'Neill builds up this function-first framework slowly there, and it'll make the differentiation in Definition 10 click once you have that foundation.

备注：内容来源于stack exchange，提问作者John