15.23: Some Mathematical Results

Before proceeding with the next section, I just want to establish few mathematical results, so that we do not get bogged down in heavy algebra later on when we should be concentrating on understanding physics.

First, if

$$\gamma=\left(1-\frac{u^{2}}{c^{2}}\right), \label{15.23.1}$$

Then, by trivial differentiation,

$$\frac{d\gamma}{du}=\frac{\gamma^{3}u}{c^{2}}. \label{15.23.2}$$

$$\dot{\gamma}=\frac{\gamma^{3}u\dot{u}}{c^{2}}. \label{15.23.3}$$

From this, we quickly find that

$$\frac{\gamma u\dot{u}}{\dot{\gamma}}=c^{2}-u^{2}. \label{15.23.4}$$

Now for a small result concerning a scalar (dot) product.

Let A be a vector such that A * A = $A^{2}$.

Then

$\frac{d}{dt}(A^{2})=2A\dot{A}$ and $\frac{d}{dt}(\bf{A\cdot A})=2A\cdot\dot{A}$

$$A\cdot\dot{A}=A\dot{A} \label{15.23.6}$$

We can now safely proceed to the next section.