Calculus: Vector Functions

Vector Functions

• A vector function maps real numbers to vectors.
• In 2D: $\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j}$, where $f(t)$ and $g(t)$ are functions of $t$.
• In 3D: $\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$.

Calculus of Vector Functions

• Differentiation of vector functions involves finding the limit: $\frac{d\vec{r}}{dt} = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}$.
• Derivative is computed component-wise: $\frac{d\vec{r}}{dt} = \frac{df}{dt}\hat{i} + \frac{dg}{dt}\hat{j} + \frac{dh}{dt}\hat{k}$.
• Integration is also component-wise: $\int \vec{r}(t) dt = \left(\int f(t) dt\right)\hat{i} + \left(\int g(t) dt\right)\hat{j} + \left(\int h(t) dt\right)\hat{k}$.

Vector Function Examples

• Example vector function: $\vec{r}(t) = t^2\hat{i} + (2t - 1)\hat{j} + \cos(t)\hat{k}$.

  • The derivative is $\frac{d\vec{r}}{dt} = 2t\hat{i} + 2\hat{j} - \sin(t)\hat{k}$.
  • The integral is $\int \vec{r}(t) dt = \left(\frac{t^3}{3} + C_1\right)\hat{i} + \left(t^2 - t + C_2\right)\hat{j} + \left(\sin(t) + C_3\right)\hat{k}$, where $C_1$, $C_2$ and $C_3$ are constants of integration.

• An object's position in space: $\vec{r}(t) = e^t\hat{i} + t^3\hat{j} + (1 - t)\hat{k}$.

• Velocity $\vec{v}(t)$ is the derivative of position $\vec{r}(t)$: $\vec{v}(t) = \frac{d\vec{r}}{dt} = e^t\hat{i} + 3t^2\hat{j} - \hat{k}$.

• Acceleration $\vec{a}(t)$ is the derivative of velocity $\vec{v}(t)$: $\vec{a}(t) = \frac{d\vec{v}}{dt} = e^t\hat{i} + 6t\hat{j} + 0\hat{k}$.

• At $t = 0$, velocity is $\vec{v}(0) = \hat{i} - \hat{k}$.

• At $t = 0$, acceleration is $\vec{a}(0) = \hat{i}$.

• Integral evaluation of $\vec{r}(t) = 2t\hat{i} + \sin(t)\hat{j}$ from $t = 0$ to $t = \pi$.

  • Integrate component-wise, then evaluate the limits.
  • $\int_0^\pi \vec{r}(t) dt = \left(\int_0^\pi 2t dt\right)\hat{i} + \left(\int_0^\pi \sin(t) dt\right)\hat{j} = \pi^2\hat{i} + 2\hat{j}$.