Differential Geometry
Theory of Curves A Few Results I Often Forget Double cross product formula $$ (\vec{a} \times \vec{b}) \times \vec{c} = \vec{b} (\vec{a} \cdot \vec{c}) - \vec{a} (\vec{b} \cdot \vec{c}) $$ Lagrange identity $$ (\vec{a} \times \vec{b}) (\vec{c} \times \vec{d})= (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})-(\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c}) $$ Suppose the vector-valued function ${a}(t)$ is nowhere zero and continuously differentiable. Then the length of ${a}(t)$ is constant if and only if: $$ \vec{a}^{\prime}(t) \cdot \vec{a}(t) \equiv 0 $$ Regular Parametric Curves A parametric curve ${\textbf{r} }(t)$ is a regular parametric curve if it satisfies the following conditions: ...