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In Jost's Riemannian Geometry and Geometric Analysis (Sect. 1.2, Chap. 1), the tangent space at a point $x_0$ in $\mathbb{R}^d$ is defined as $$T_{x_0}\mathbb R^d=\{x_0\}\times E$$ where $E$ is the vector space spanned by $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$. Then the books says: "Here, $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$ are the partial derivatives at the point $x_0$." This is where I get confused. They are the partial derivatives of what? The only partial derivative I know is that of a function, but no function is given here.

Sure, if one wants to argue that $\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^d}$ are just formal notations here that don't mean anything other than a formal basis of $E$, then I can accept that even though I have doubts. But then there comes something that confuses me even more. If $f:\mathbb R^d\to\mathbb R^c$ is a differentiable map, then the derivative of $f$ at $x_0$ is defined to be (Einstein convention is used below) $$df(x_0):T_{x_0}\mathbb R^d\to T_{x_0}\mathbb{R}^c\\ \quad v^i\frac{\partial}{\partial x^i}\mapsto v^i\frac{\partial f^j}{\partial x^i}\frac{\partial}{\partial f^i}$$ So apparently $\frac{\partial}{\partial f^j}$ here depend on $f$ and are not arbitraily selected, so the notation cannot simply be a formal one, which brings me back to the original question: what does $\frac{\partial}{\partial x^i}$ and $\frac{\partial}{\partial f^j}$ mean?

trisct
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2 Answers2

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The tangent space $T_pM$ can be seen as the space of local linear operators acting on the functions $f: M \rightarrow \mathbb R$. If you have vector $v\in T_pM$ you can define how it acts on a function: $$ v(f) = \left.\frac{d f(\gamma_v(t))}{dt}\right|_{t=0} $$ where $\gamma_v$ is any curve on $M$ such that $\gamma_v(0) = p$ and $\frac{d\gamma_v}{dt}(0) = v$.

Given a coordinate system $(x_i)$ you can find that there exist vectors in $T_pM$ that act on functions exactly like the partial derivatives $\frac{\partial}{\partial x_i}$, that is $v_i(f) = \frac{\partial f}{\partial x_i}(p)$. They are therefore denoted $v_i = \frac{\partial}{\partial x_i}$. Such vectors form a basis of $T_pM$, so any vector can be written as $$ v = v^i \frac{\partial}{\partial x_i} $$

If the point of differentiation is obvious, the vetor can be denoted as $v_i=\left.\frac{\partial}{\partial x_i}\right|_p$. Sometimes $\frac{\partial}{\partial x_i}$ can also denote the whole vector field, defining a vector at every point of the manifold.

Adam Latosiński
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The notation $\frac{\partial f}{\partial x_j}$ means: take $f$, which is a function on the manifold $M$, consider its expression in the coordinate system $(x_1, x_2, \ldots, x_n)$, so that $f$ is now a function on $\mathbb R^n$, take the partial derivative with respect to the $j$-th variable.

The part in bold is often not said explicitly and that can become confusing.

Giuseppe Negro
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  • I know what $\frac{\partial f}{\partial x_j}$ means, it is a number. But now I am curious about what $\frac{\partial}{\partial x_j}$ means, and apparently it is not a number nor a operator in my context. – trisct Jul 05 '19 at 08:41
  • It is a linear operator, defined on the space of smooth functions on the manifold. – Giuseppe Negro Jul 05 '19 at 08:45
  • So $E$ is a vector space formally generated by these linear opeartors? – trisct Jul 05 '19 at 08:47
  • Yes, but it is not just formal. E is a vector space spanned by these operators. – Giuseppe Negro Jul 05 '19 at 08:48
  • @GiuseppeNegro So if $f$ is a function on the manifold and $X$ is a chart, are you saying that $\partial / \partial x_j$ takes $f$ as an input, or $f \circ X$ as an input? – user56202 Oct 04 '23 at 23:17
  • @user56202: According to the definition I wrote above, it is the first alternative. Of course in many practical situations one may abuse notations a little, and apply that to $f\circ X$, but typically in these cases it is always clear from the context what exactly is going on. – Giuseppe Negro Oct 04 '23 at 23:21
  • @GiuseppeNegro Thank you, I think I understand. So $\partial / \partial x_j$ IS a tangent vector itself, and it is dependent on the chart $X$, so we could perhaps include the $X$ somewhere in that symbol to indicate that. But for simplicity, we do not put the $X$ in the symbol. Is this right? – user56202 Oct 04 '23 at 23:30
  • @user56202: Look, we are discussing notation here. Don't make the rookie mistake to think that notation is something immutable, that somebody above decides once and for all. Notations depend on the context. So, coming to your last question, there is no "right" or "wrong". What you wrote is plausible and definitely would make sense in a lot of mathematical writings, but don't believe it's the absolute truth, for somebody else may use the symbols with different meanings in mind. – Giuseppe Negro Oct 05 '23 at 22:24
  • @GiuseppeNegro Thanks, that is a good perspective. – user56202 Oct 06 '23 at 02:49