1. Differentiation of vectors
1. basis vectors
\mathbf{\hat{e}}_{\rho}=\cos\phi\mathbf{i}+\sin\phi\mathbf{j},\\\mathbf{\hat{e}}_{\phi}=-\sin\phi\mathbf{i}+\cos\phi\mathbf{j}\\
\frac{d\mathbf{\hat{e}}_{\rho}}{dt}=-\sin\phi\frac{d\phi}{dt}\mathbf{i}+\cos\phi\frac{d\phi}{dt}\mathbf{j}=\dot{\phi}\mathbf{\hat{e}}_{\phi},\\\frac{d\mathbf{\hat{e}}_{\phi}}{dt}=-\cos\phi\frac{d\phi}{dt}\mathbf{i}-\sin\phi\frac{d\phi}{dt}\mathbf{j}=-\dot{\phi}\mathbf{\hat{e}}_{\rho}
2. Composite vectors
\begin{gathered}
\frac{d}{du}(\phi\mathbf{a})=\phi\frac{d\mathbf{a}}{du}+\frac{d\phi}{du}\mathbf{a}, \\
\frac{d}{du}(\mathbf{a}\cdot\mathbf{b})=\mathbf{a}\cdot\frac{d\mathbf{b}}{du}+\frac{d\mathbf{a}}{du}\cdot\mathbf{b}, \\
\frac{d}{du}(\mathbf{a}\times\mathbf{b})=\mathbf{a}\times\frac{d\mathbf{b}}{du}+\frac{d\mathbf{a}}{du}\times\mathbf{b},\\
\frac{d\mathbf{a}(s)}{du}=\frac{ds}{du}\frac{d\mathbf{a}}{ds}.
\end{gathered} \\
Special example:
A vector with constant magnitude is always perpendicular to \frac{d\mathbf{a}}{du}.
\frac{d}{du}(\mathbf{a}\cdot\mathbf{a})=2\mathbf{a}\cdot\frac{d\mathbf{a}}{du}=\frac{d}{du}(a^2)=0
3. Integration of vectors
\int\mathbf{a}(u)du=\mathbf{A}(u)+\mathbf{b}\\
\int_{u_1}^{u_2}\mathbf{a}(u)du=\mathbf{A}(u_2)-\mathbf{A}(u_1)
Example:
If m\frac{d^2\mathbf{r}}{dt^2}=-\frac{GMm}{r^2}\hat{\mathbf{r}}, then prove that \mathbf{r}\times d\mathbf{r}/dt = constant. (Page339)
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