Lecture 5: Tangent Spaces (International Winter School on Gravity and Light 2015)
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when someone is confused about how to differentiate a function on a manifold, what a tangent vector really is, or why the partial derivative notation on charts looks the way it does.
Recommended by 4 people across 5 different months, 2017 to 2022.
What it is
Lecture 5 of Frederic Schuller's 24-part general relativity course at the 2015 International Winter School on Gravity and Light. It builds the tangent space of a smooth manifold from the velocity of curves, then covers coordinate basis vectors, chart notation and the differential df.
What people said
Strong rec RozaTh · 2017-11 · Mathematics Stack Exchange · on “Silly question about Directional Derivatives.”
The velocity of a curve on a manifold is a function that eats a function on the manifold and gives back the directional derivative of the function along the curve. One of the best explanations is given here: [this video]
Strong rec peek-a-boo · 2019-10 · Mathematics Stack Exchange · on “(Algebraic) derivations $\frac{\partial}{\partial x_j}$ are linearly independent”
I highly recommed you watch this lecture [this video] by Dr Frederic Schuller on tangent spaces (in particular minute 53, but watch the entire thing :))
Recommends peek-a-boo · 2020-07 · Mathematics Stack Exchange · on “Upper and lower indices when identifying tangent space vectors with underlying vector space”
@MrArsGravis Sure, if you want to express it in that way, but I personally find that very confusing terminology, which is very easily misunderstood. $\frac{\partial}{\partial x^i}(p)$ means every smooth function $f$ gets mapped to the real number $ \partial_i(f\circ x^{-1}) [LINK] )$. Which means you take the $i^{th}$ partial derivative of the chart representative function $f\circ x^{-1}$, and then evaluate at the point $x(p)$. I highly recommend you watch this lecture [this video] , (especially 50 minutes onwards)
Recommends Clemens Bartholdy · 2021-05 · Mathematics Stack Exchange · on “How exactly does it make sense to differentiate a function whose input is a point on a manifold?”
In this lecture [this video] , prof Frederic Scheuller introduces the concept of the tangent vector space defined the following way by introducing the concept of velocity at a point: Let $(M,\theta,A)$ be a smooth manifold, and a curve $\gamma: R \to M$ at least $C^1$ (One differentiable and derivative is continous). Suppose $\gamma(\lambda_o)=p$ , then the velocity at $p$ is defined as: $v_{\gamma,p} : C^{\infty} (M) \to R$ Where $C^{\infty}(M) := \{ f: M \to R | f \text{ is a smooth function} \}$ equipped with an addition of the form $$(f+g)(p)= f(p) + g(p)$$ and s multiplication of the form $$( \lambda \cdot g)(p) = \lambda g(p)$$ where the addition and scaling is in the same sense as operatio
Recommends Br1nd4 · 2022-01 · Mathematics Stack Exchange · on “Is my proof that we can endow the tangent space with a vector space structure alright?”
First some initial definitions (because I am working from a physics point of view rather than from a mathematician's point of view, and I suspect the way this problem is approached will vary slightly between the two disciplines): $\gamma:\mathbb{R}\supseteq U\to M$ is a curve on $M$ if $\gamma$ is differentiable. The velocity , $v_{\gamma,p}:C^{\infty}(M)\to\mathbb{R}$ is a linear function defined as $v_{\gamma,p}=d_{\lambda}(f\circ\gamma)|_{\lambda_0}$ , where $\gamma$ is a function in $\lambda$ and $p=\gamma(\lambda_0)$ . Note $d_\lambda=\frac{d}{d\lambda}$ . I am trying to prove that the tangent space at a point $p$ on a manifold $M$ defined by $$T_p(M)=\{v_{\gamma,p}:\gamma\text{ is a cu
Cites peek-a-boo · 2020-06 · Mathematics Stack Exchange · on “Clarifying the chain rule terminology in differential geometry calculuations”
If you're having trouble with the notation in differential geometry, my suggestion is to completely avoid Leibniz's notation (temporarily) and write out everything in completely precise notation, being careful with what the function is vs where it is being evaluated. Let's first study what happens in $\Bbb{R}^n$ before moving on to the manifold case. Consider a differentiable map $F: \Bbb{R}^n \to \Bbb{R}$ and a differentiable curve $\lambda: \Bbb{R} \to \Bbb{R}^n$ (of course you can restrict everything to open subsets but I don't feel like introducing too many new letters for new domains). We can now form the composite function $F \circ \lambda:\Bbb{R} \to \Bbb{R}$ , and ask what is the der
Cites FFjet · 2021-01 · Mathematics Stack Exchange · on “Question about $\partial_i \left(f \circ x^{-1}\right)(x(p))$”
In this video [this video] (at about 55:00), the lecturer wrote $\partial_i \left(f \circ x^{-1}\right)(x(p))$ as $\left (\dfrac{\partial f}{\partial x^{i}} \right)_p$ . I am very confused about this notation. The lecturer says its not a partial derivative, but how is it useful to just denote like this? $x$ is a function in $\partial_i \left(f \circ x^{-1}\right)(x(p))$ but how can it be like a coordinate in $\left (\dfrac{\partial f}{\partial x^{i}} \right)_p$ ? Does it have something to do with partial derivatives? I would appreciate it if there's someone could clarify this for me.
Cites Clemens Bartholdy · 2021-05 · Mathematics Stack Exchange · on “Does df mean a differential or gradient?”
In wikipedia [LINK] and also how I had originally learned, $df$ is said to be the differential of a scalar function $f$ and is a linear map from $R^n \to R$ . However, in this lecture prof. Frederic Schuller mentions that the gradient of $f$ is defined as a linear map from $T_p M \to R$ (see 1:27:08 of this video [this video] ). So, what really is df?
Cites Clemens Bartholdy · 2021-05 · Mathematics Stack Exchange · on “How exactly does it make sense to differentiate a function whose input is a point on a manifold?”
I don't think this answer was what the professor was intending, at 55:57 [this video] the professor states that this is NOT supposed to be the partial derivative