https://keqing996.github.io/categories/differentialgeometry/ Recent content in DifferentialGeometry on BlueHour Hugo en Thu, 07 Oct 2021 00:00:00 +0000 https://keqing996.github.io/posts/math/differentialgeometry/20211007_differentialgeometry/ Thu, 07 Oct 2021 00:00:00 +0000 https://keqing996.github.io/posts/math/differentialgeometry/20211007_differentialgeometry/ <h1 id="theory-of-curves">Theory of Curves</h1> <h2 id="a-few-results-i-often-forget">A Few Results I Often Forget</h2> <ol> <li>Double cross product formula</li> </ol> <div class="math-display"> $$ (\vec{a} \times \vec{b}) \times \vec{c} = \vec{b} (\vec{a} \cdot \vec{c}) - \vec{a} (\vec{b} \cdot \vec{c}) $$ </div><ol start="2"> <li>Lagrange identity</li> </ol> <div class="math-display"> $$ (\vec{a} \times \vec{b}) (\vec{c} \times \vec{d})= (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})-(\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c}) $$ </div><ol start="3"> <li>Suppose the vector-valued function <span class="math-inline">${a}(t)$</span> is nowhere zero and continuously differentiable. Then the length of <span class="math-inline">${a}(t)$</span> is constant if and only if:</li> </ol> <div class="math-display"> $$ \vec{a}^{\prime}(t) \cdot \vec{a}(t) \equiv 0 $$ </div><h2 id="regular-parametric-curves">Regular Parametric Curves</h2> <p>A parametric curve <span class="math-inline">${\textbf{r} }(t)$</span> is a regular parametric curve if it satisfies the following conditions:</p>