I'm currently watching these Lectures on General Relativity and I tried to work out a simple example to help me understand the content of the lectures better. I tried to calculate the acceleration of a particle moving in a uniform circular motion, but using the "new" formalism introduced in the lectures.
I choose my manifold $\mathcal{M}$ to be $\mathbb{R}^{2}$. I define the curve $\gamma:\mathbb{R}\rightarrow \mathcal{M}$ where:
\begin{equation} \gamma(\lambda)=(\cos\lambda,\sin\lambda) \end{equation}
My Chart is Simply $x^1(a^1,a^2)=a^1$ and $x^2(a^1,a^2)=a^2$. The first thing I do is calculate the velocity, in the way defined in the lectures. According to the lectures the tangent vectors are maps of the form: \begin{equation} v_{\gamma, p}=\dot{\gamma}_{x}^{i}(\lambda) \cdot\left(\frac{\partial}{\partial x^{i}}\right)_{p} \end{equation} Where \begin{equation} \dot{\gamma}_{x}^{i}(\lambda)=\left(x^{i} \circ \gamma\right)^{\prime} \end{equation} and \begin{equation} \left(\frac{\partial}{\partial x^{i}}\right)_{p}(f)=\left.\partial_{i}\left(f \circ x^{-1}\right)\right|_{x(p)} \end{equation} Where $\frac{\partial}{\partial x^{i}}$ is a basis vector and $f:\mathcal{M}\rightarrow \mathbb{R}$ a scalar function. Note that this far in the lectures that are vectors are defined only as maps that act on these kinds of scalar functions. Therefore my vector field is the following: \begin{equation} X=-\sin\lambda\frac{\partial}{\partial x^{1}}+\cos\lambda\frac{\partial}{\partial x^{2}}=X^i\frac{\partial}{\partial x^{i}} \end{equation} Where $X^i=\left(x^{i} \circ \gamma\right)^{\prime}$ this is a function from $X^i:\mathbb{R}\rightarrow \mathbb{R}$ and this is what confuses me. Later in the lecture he generalizes the covariant derivative to vector fields (at minute $32$).I apply this to my vector field and I get the following: \begin{equation} \begin{aligned} \nabla_{X} X &=\nabla_{X^{i}\frac{\partial}{\partial x^{i}}} \left(X^{j} \frac{\partial}{\partial x^{j}}\right) \\ &=X^{i} \nabla_{\frac{\partial}{\partial x^{i}}}\left(X^{j} \frac{\partial}{\partial x^{j}}\right) \\ &=X^{i}\left(\nabla_{\frac{\partial}{\partial x^{i}}} X^{j}\right) \frac{\partial}{\partial x^{j}}+X^{i} X^{j}\left(\nabla_{\frac{\partial}{\partial x^{i}}} \frac{\partial}{\partial x^{j}}\right)\\ &=X^{i}\frac{\partial}{\partial x^{i}}\left( X^j \right) \frac{\partial}{\partial x^{j}}+X^{i} X^{j}\left(\nabla_{\frac{\partial}{\partial x^{i}}} \frac{\partial}{\partial x^{j}}\right) \end{aligned} \end{equation} The first derivative on the right hand side according to the definition earlier is this: \begin{equation} \left(\frac{\partial}{\partial x^{i}}\right)_{p}(X^j)=\left.\partial_{i}\left(X^j \circ x^{-1}\right)\right|_{x(p)} \end{equation} This expression makes no sense to me because the maps don't fit, $x^{-1}:\mathbb{R}^2\rightarrow \mathcal{M}$ and $X^i:\mathbb{R}\rightarrow \mathbb{R}$. Can someone help me find where I'm making the mistake here?