Differentiation Rules for Curve Parameterisations

Theorem: Linearity of Differentiation

The differentiation operation for curve parameterisations is linear - for all curve parameterisations $\mathbf{u},\mathbf{v}:I\subseteq \mathbb{R}\to {\mathbb{R}}^n$ and all $\mu ,\lambda \in \mathbb{R}$:

$\frac{\mathrm{d}}{\mathrm{d}t}[\lambda \mathbf{u}(t)+\mu \mathbf{v}(t)]=\lambda {\mathbf{u}}^{\prime }(t)+\mu {\mathbf{v}}^{\prime }(t)$

PROOF

TODO

Theorem: Chain Rule

For every differentiable real function $f:I\subseteq \mathbb{R}\to \mathbb{R}$ and every differentiable curve parameterisation $\mathbf{r}:f(I)\to {\mathbb{R}}^n$:

$\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{r}(f(t))=f^{\prime }(t){\mathbf{r}}^{\prime }(f(t))$

PROOF

TODO

Theorem: Product Rules

• Basic Product Rule: For every differentiable curve parameterisation $\mathbf{r}:I\to {\mathbb{R}}^n$ and every differentiable real function $f:I\subseteq \mathbb{R}\to \mathbb{R}$

$\frac{\mathrm{d}}{\mathrm{d}t}[f(t)\mathbf{r}(t)]=f^{\prime }(t)\mathbf{r}(t)+{\mathbf{r}}^{\prime }(t)f(t)$

• Dot Product Rule: For the dot product of all differentiable curve parameterisations $\mathbf{u},\mathbf{v}:I\subseteq \mathbb{R}\to {\mathbb{R}}^n$

$\frac{\mathrm{d}}{\mathrm{d}t}[\mathbf{u}(t)\cdot \mathbf{v}(t)]={\mathbf{u}}^{\prime }(t)\cdot \mathbf{v}(t)+\mathbf{u}(t)\cdot {\mathbf{v}}^{\prime }(t)$

• Cross Product Rule: For the cross product of all differentiable curve parameterisations $\mathbf{u},\mathbf{v}:I\subseteq \mathbb{R}\to {\mathbb{R}}^3$

$\frac{\mathrm{d}}{\mathrm{d}t}[\mathbf{u}(t)\times \mathbf{v}(t)]={\mathbf{u}}^{\prime }(t)\times \mathbf{v}(t)+\mathbf{u}(t)\times {\mathbf{v}}^{\prime }(t)$

PROOF

TODO