Derivative of Vector Function

Definition

Suppose a vector function $\mathbf{r}(t)=⟨f(t),g(t),h(t)⟩:I\to {\mathbb{R}}^3$ ${\mathbf{r}}^{\prime }(t)=\frac{d\mathbf{r}}{dt}=⟨\frac{df}{dt},\frac{dg}{dt},\frac{dh}{dt}⟩=\frac{df}{dt}\mathbf{i}+\frac{dg}{dt}\mathbf{j}+\frac{dh}{dt}\mathbf{k}$

Properties

Rules of Computation

Let $\mathbf{u}$ and $\mathbf{v}$ be differentiable vector functions of $t$, $a,b$ any scalars, and $f$ any differentiable scalar functions.

• Linearity: $(a\mathbf{u}\pm b\mathbf{v}{)}^{\prime }(t)=a{\mathbf{u}}^{\prime }(t)\pm b{\mathbf{v}}^{\prime }(t)$
• Dot product rule: $(\mathbf{u}\cdot \mathbf{v}{)}^{\prime }(t)={\mathbf{u}}^{\prime }(t)\cdot \mathbf{v}(t)+\mathbf{u}(t)\cdot {\mathbf{v}}^{\prime }(t)$
• Cross product rule: $(\mathbf{u}\times \mathbf{v}{)}^{\prime }(t)={\mathbf{u}}^{\prime }(t)\times \mathbf{v}(t)+\mathbf{u}(t)\times {\mathbf{v}}^{\prime }(t)$
• Chain rune: $(\mathbf{u}(f(t)){)}^{\prime }=f^{\prime }(t){\mathbf{u}}^{\prime }(f(t))$

Facts

Vector Function of Constant Length

If $\mathbf{r}$ is a differentiable vector function of $t$ of constant length, then $\mathbf{r}\cdot \frac{d\mathbf{r}}{dt}=0$