Section 3.2 - Differentiation and Integration of Vector-Valued Functions

Section Objectives

1. Evaluate and use the derivative of a vector-valued function.
2. Compute the unit tangent vector for a vector-valued function.
3. Integrate vector-valued functions.

Differentiation

The derivative of a vector-valued function is defined by a limit,

just like the derivative you saw in Calculus I. However, since we evaluate limits of vector-valued functions component by component, it is probably not surprising to you that we can evaluate derivatives in a similar way:

Examples

• Let $\vec{r}(t)=3\cos t\, \hat{\imath} + 4\sin t \, \hat{\jmath}$. Compute $\vec{r}'(t)$.

• Let $\vec{r}(t)=e^t\sin t\, \hat{\imath} + e^t \cos t \, \hat{\jmath} - e^{2t} \, \hat{k}$. Compute $\vec{r}'(t)$.

• Let $\vec{r}(t)=\sin 3t\, \hat{\imath} - \cos 3t \, \hat{\jmath} +4 \, \hat{k}$. Compute $\vec{r}(t) \cdot \vec{r}(t)$ and its derivative.

Properties of the derivative

Suppose $\vec{r}(t)$, $\vec{u}(t)$ and $f(t)$ are differentiable functions of $t$, and further suppose that $c$ is a constant.

1. $\displaystyle \frac{d}{dt} [c\vec{r}(t)] = c \vec{r}'(t)$
2. $\displaystyle \frac{d}{dt} [\vec{r}(t) \pm \vec{u}(t)] = \vec{r}'(t) \pm \vec{u}'(t)$
3. $\displaystyle \frac{d}{dt} [f(t) \, \vec{r}(t)] = f'(t) \, \vec{r}(t) + f(t) \, \vec{r}'(t)$
4. $\displaystyle \frac{d}{dt} [\vec{r}(t) \cdot \vec{u}(t)] = \vec{r}'(t) \cdot \vec{u}(t) + \vec{r}(t) \cdot \vec{u}'(t)$
5. $\displaystyle \frac{d}{dt} [\vec{r}(t) \times \vec{u}(t)] = \vec{r}'(t) \times \vec{u}(t) + \vec{r}(t) \times \vec{u}'(t)$
6. $\displaystyle \frac{d}{dt} \, \vec{r}(f(t)) = \vec{r}'(f(t)) \, f'(t)$
7. If $\displaystyle \vec{r}(t) \cdot \vec{r}(t) = c$, then $\vec{r}(t) \cdot \vec{r}'(t)=0$. (As a consequence, vector-valued functions of constant magnitude are orthogonal to their derivatives.)

Each of these properties is easy to prove. Unfortunately, because we will usually compute our derivatives component by component, we will not find these properties very useful. Property 7 is an exception---we will use this property often.

Examples

• Let $\vec{r}(t)=2\cos 5t \, \hat{\imath} - 2 \sin 5t \, \hat{\jmath} +4t \, \hat{k}$. Compute $\vec{r}'(t)$, $\|\vec{r}'(t)\|$, $\vec{r}''(t)$, and $\vec{r}'(t) \cdot \vec{r}''(t)$.

Tangent vectors

Recall that the derivative of a scalar-valued function gives the slope of the tangent line. For vector-valued functions, the derivative gives a tangent vector.

In the graph above, the tangent vector shown is in fact a unit vector. It is a unit tangent vector.

Suppose $C$ is a curve in space defined by the vector-valued function $\vec{r}$. Assume $\vec{r}'(t)$ exists and is not the zero vector. Then the principal unit tangent vector at $t$ is given by

If we think of $\vec{r}(t)$ as giving the position of a particle at time $t$, then $\hat{T}(t)$ is the unit vector that points in the direction of motion.

Examples

• Let $\vec{r}(t)=2\cos 5t \, \hat{\imath} - 2 \sin 5t \, \hat{\jmath} +4t \, \hat{k}$. Compute $\hat{T}(t)$.

• Let $\vec{r}(t)=(t^2+1) \, \hat{\imath} - (3t^2+t)\, \hat{\jmath}\, + (3t-7) \, \hat{k}$. Compute $\hat{T}(t)$.

• Suppose $C$ is a curve in space defined by the vector-valued function $\vec{r}$. We say that $C$ is smooth on the open interval $I$ if $\vec{r}'(t)$ is continuous and never zero on $I$. In the examples above, are the curves smooth? Notice that the principal unit tangent vector always exists for smooth curves.

Integration

As with limits and derivatives, we evaluate the integral of a vector-valued function component by component. That is, the integral of a vector-valued function is obtained by integrating each component:

For indefinite integrals, it is very important to include a scalar constant of integration with each component. Alternatively, you can include a vector constant of integration for the entire integral.

Examples

• Let $\vec{r}(t)=(t^2-3t+4) \, \hat{\imath} + (4t+3) \, \hat{\jmath}$. Evaluate $\displaystyle \int_0^2 \vec{r}(t) \, dt$.

• Let $\displaystyle \vec{r}(t)=\frac{2}{t+1} \, \hat{\imath} + \frac{t}{t^2+1} \, \hat{\jmath} + \frac{1}{t^2+1} \, \hat{k}$. Evaluate $\displaystyle \int \vec{r}(t) \, dt$.

• Find $\vec{r}(t)$ if $\vec{r}'(t)= \displaystyle \cos(2t) \, \hat{\imath} -2 \sin(t) \, \hat{\jmath} + \frac{1}{t^2+1} \, \hat{k}$ and $\vec{r}(0)=3 \hat{\imath} -2 \hat{\jmath} + \hat{k}$.