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15.23: Some Mathematical Results

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  • Before proceeding with the next section, I just want to establish few mathematical results, so that we don’t 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 = \( A^{2}\).


    \( \frac{d}{dt}(A^{2})=2A\dot{A}\) 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.