Compute Christoffel symbols of a rotating cartesian coordinate system

Question

Suppose we have a smooth manifold $(\mathbb{R}^3, \mathcal{O}_{\mathbb{R}^4}, \{(\mathbb{R}^3,x),(\mathbb{R}^3,y)\},\nabla,t)$ where $t:\mathbb{R}^3\rightarrow\mathbb{R}$ is such that $t(a,b,c,d)=a$, $x=Id_{\mathbb{R}^3}$, $\nabla$ is such that in the chart $(\mathbb{R}^3,x)$ the connection coefficients ${\Gamma_{(x)}} ^{a}_{bc} =0$ and $y$ is defined through the identities $$t=x^0=y^0$$ $$x^1=y^1\cos\theta-y^2\sin\theta$$ $$x^2=y^1\sin\theta+y^2\cos\theta$$ for some $\theta=\omega t$ where $\omega\in\mathbb{R}$. This structure attempts to model a three dimensional Newtonian spacetime with an inertial chart and to identify which coefficients of the connection in the non-inertial chart are different than zero. In my computation of the connection coefficients I realized my understanding of differential geometry is very bad. I found $\frac{\partial x^1}{\partial y^0}=0$ and $\frac{\partial x^1}{\partial y^1}=\cos\theta$ which implies $$\frac{\partial^2 x^1}{\partial y^1 \partial y^0}=0\neq-\omega\sin\theta=\frac{\partial^2 x^1}{\partial y^0\partial y^1}$$ Why don't my partial derivatives commute? Any extra help on calculating the coefficients is greatly appreciate. If anybody knows of any textbooks that give a formal treatment of Newtonian space time, the information would be very helpful!

Answer 1

It seems that you forgot that $\theta$ depends on $y^0$ when you computed $\dfrac{\partial x^1}{\partial y^0}$. The correct answer should be $$\frac{\partial x^1}{\partial y^0} = -\omega(y^1\sin\theta+y^2\cos\theta),$$ and then $$\frac{\partial^2 x^1}{\partial y^1\partial y^0} = -\omega\sin\theta,$$ as you desired.