Is this statement acceptable? (time derivative of a rotating vector)

[Clockclocle]

Homework Statement

Is this statement acceptable?

Relevant Equations

..........

I understand the approximation statement but he divide the |delta t| in the left but only delta t on the right. Is it true because delta phi would have the same sign as delta t ?

 

[kuruman]

Yes. By definition $\Delta \phi$ is positive and the angle increases in the direction of the arrow in the top figure. $\Delta t$ is always positive.

 

[Clockclocle]

Yes. By definition $\Delta \phi$ is positive and the angle increases in the direction of the arrow in the top figure. $\Delta t$ is always positive.

He also use the fact that lim |$\Delta A$/$\Delta t$| = |lim $\Delta A$/$\Delta t$|. Is that accepted?

 

[kuruman]

There is a more formal way to show the same thing which I prefer.

You have a vector $\mathbf A$ that has constant magnitude $A$ and direction $\mathbf {\hat a}$ that changes with time. You can write the vector as its constant magnitude times the unit vector specifying the direction, $\mathbf A=A~\mathbf {\hat a}$. Now $$\frac{d\mathbf A}{dt}=A\frac{d\mathbf {\hat a}}{dt}.$$ To find the derivative of the unit vector $\mathbf a$, consider the drawing on the right. A

unit vector in the direction of changing angle $\theta$ is perpendicular to $\mathbf {\hat a}.$ In terms of the fixed Cartesian unit vectors
$$\begin{align} & \mathbf {\hat a}=\cos\!\theta~\mathbf {\hat x}+\sin\!\theta~\mathbf {\hat y} \\
& \mathbf {\hat {\theta}}=-\sin\!\theta~\mathbf {\hat x}+\cos\!\theta~\mathbf {\hat y}
\end{align}$$Now from equation (1) $$\frac{d\mathbf {\hat a}}{dt}=\frac{d\theta}{dt}(-\sin\!\theta~\mathbf {\hat x}+\cos\!\theta~\mathbf {\hat y})=\frac{d\theta}{dt}\mathbf {\hat{\theta}}$$ and the time rate of change of the constant-magnitude vector $\mathbf A$ is $$\frac{d\mathbf A}{dt}=A\frac{d\mathbf {\hat a}}{dt}=A\frac{d\theta}{dt}\mathbf {\hat{\theta}}.$$

 

[haruspex]

He also use the fact that lim |$\Delta A$/$\Delta t$| = |lim $\Delta A$/$\Delta t$|. Is that accepted?

Not in general. Consider $\lim _{x\rightarrow 0}(-1)^{\lfloor \frac 1x\rfloor}$.