- #1
jk22
- 729
- 24
Suppose we have to deal with the question : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=?\frac{\partial}{\partial y}\frac{\partial}{\partial x}$$
This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the case, since : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=\frac{d}{\dot{f}dt}\frac{d}{\dot{g}dt}=\frac{d^2}{\dot{f}\dot{g}dt^2}-\frac{\ddot{g}d}{\dot{f}\dot{g}^2 dt}\neq\frac{\partial}{\partial y}\frac{\partial}{\partial x}=\frac{d}{\dot{g}dt}\frac{d}{\dot{f}dt}=\frac{d^2}{\dot{f}\dot{g}dt^2}-\frac{\ddot{f}d}{\dot{g}\dot{f}^2 dt}$$.
Is this equal to the covariant derivative ?
For example can we then say that if we consider a curve on a sphere that those partial derivatives do not commute in general ?
This seems true for independent variables. But if at the end x and y are linked in some way like $$x=f(t),y=g(t)$$ this is no more the case, since : $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}=\frac{d}{\dot{f}dt}\frac{d}{\dot{g}dt}=\frac{d^2}{\dot{f}\dot{g}dt^2}-\frac{\ddot{g}d}{\dot{f}\dot{g}^2 dt}\neq\frac{\partial}{\partial y}\frac{\partial}{\partial x}=\frac{d}{\dot{g}dt}\frac{d}{\dot{f}dt}=\frac{d^2}{\dot{f}\dot{g}dt^2}-\frac{\ddot{f}d}{\dot{g}\dot{f}^2 dt}$$.
Is this equal to the covariant derivative ?
For example can we then say that if we consider a curve on a sphere that those partial derivatives do not commute in general ?