- #1
PFuser1232
- 479
- 20
The magnitude of the parallel component of the time derivative of a vector ##\vec{A}## is given by:
$$|\frac{d\vec{A}_{\parallel}}{dt}| = |\frac{dA}{dt}|$$
Where ##A## is the magnitude of the vector.
Can we write the actual derivative in vector form as ##\frac{dA}{dt} \hat{A}##? Notice how I dropped the absolute value sign.
The reason I'm asking this is when I tried deriving acceleration in polar coordinates, I split both ##\Delta \vec{v}_r## and ##\Delta \vec{v}_{θ}## into components parallel and perpendicular to each of the vectors, then I divided each of the terms by ##\Delta t## and took the limit as ##\Delta t →0##. I ran into a problem, though. The scalars ##v_r## and ##v_{θ}## do not represent magnitudes, but scalar components that can be positive, negative, or zero. So, for the term ##\frac{d\vec{v}_{r\parallel}}{dt}##, I wrote ##\frac{d|v_r|}{dt} \hat{v}_r## where ##\hat{v}_r## is a unit vector always pointing in the direction of ##\vec{v}_r##. I then realized that if ##v_r## is positive, ##\hat{v}_r## and ##\hat{r}## point in the same direction. If ##v_r## is negative, ##\hat{v}_r## and ##\hat{r}## point in opposite directions, giving the same result in both cases, namely ##\frac{dv_r}{dt} \hat{r}##. Is this correct?
My book does not place emphasis on the fact that ##v_r## may be negative and arrives at the desired result much quicker.
$$|\frac{d\vec{A}_{\parallel}}{dt}| = |\frac{dA}{dt}|$$
Where ##A## is the magnitude of the vector.
Can we write the actual derivative in vector form as ##\frac{dA}{dt} \hat{A}##? Notice how I dropped the absolute value sign.
The reason I'm asking this is when I tried deriving acceleration in polar coordinates, I split both ##\Delta \vec{v}_r## and ##\Delta \vec{v}_{θ}## into components parallel and perpendicular to each of the vectors, then I divided each of the terms by ##\Delta t## and took the limit as ##\Delta t →0##. I ran into a problem, though. The scalars ##v_r## and ##v_{θ}## do not represent magnitudes, but scalar components that can be positive, negative, or zero. So, for the term ##\frac{d\vec{v}_{r\parallel}}{dt}##, I wrote ##\frac{d|v_r|}{dt} \hat{v}_r## where ##\hat{v}_r## is a unit vector always pointing in the direction of ##\vec{v}_r##. I then realized that if ##v_r## is positive, ##\hat{v}_r## and ##\hat{r}## point in the same direction. If ##v_r## is negative, ##\hat{v}_r## and ##\hat{r}## point in opposite directions, giving the same result in both cases, namely ##\frac{dv_r}{dt} \hat{r}##. Is this correct?
My book does not place emphasis on the fact that ##v_r## may be negative and arrives at the desired result much quicker.