Content Questions

Is $\nabla = (\frac{\partial}{\partial x} \hat{x} + \frac{\partial}{\partial y} \hat{y} + \frac{\partial}{\partial z} \hat{z})$ ?

How did you get from $-\int_1^2 (\nabla U) \cdot d\vec{r} = - \int_1^2 dU$ ?

In $\frac{dU}{dt} = \sum_i \frac{\partial U}{\partial x_i}\frac{\partial x_i}{\partial t} + \frac{\partial U}{\partial t}$ , does the second term come from the chain rule?

When proving conservation of energy how did we turn $\nabla U\cdot \dot{\vec{r}}$ into $-\vec{F}\cdot \dot{\vec{r}}$ ?

Where did $t-t_0 = \pm \int_{x_0}^x \frac{dx'}{\sqrt{\frac{2}{m}(E-U(x'))}}$ ? come from?

In $t-t_0 = \pm \int_{x_0}^x \frac{dx'}{\sqrt{\frac{2}{m}(E-U(x'))}}$ , how do you get

If you integrate $\pm \frac{dx}{\sqrt{\frac{2}{m}(E-U(x))}}$ to solve for