7. Calculus of Vector Valued Functions

We have defined curves in space as vector valued functions. Calculus wants to know about the rate of change along such a curve. Today, we learn about both derivatives and integrals of vector valued functions:

Derivatives of Vector Valued Functions

To find the derivative of $\large \vec{r}(t)=\left<x(t),y(t),z(t)\right>$, just take the derivative of each component:

$\large \vec{r'}(t)=\left<x'(t),y'(t),z'(t)\right>$

The derivative is a vector, which we call the tangent vector. Here's an illustration of the tangent vector (the blue vector is the position vector, and the black vector is the tangent):

tangentVector

With the tangent vector as a direction vector, we can find the equation of a line tangent to our curve. Just like in single variable calculus, tangents are important because they allow us to approximate the value of the function near the point of tangency.

The unit tangent vector is, not surprisingly, a vector of length one in the same direction as the tangent vector:

$$\large \hat{T}(t)=\frac{\vec{r'}(t)}{\left|\vec{r'}(t)\right|}$$

Product Rules

There are three product rules for derivatives of vector valued functions, and they all work just like you would want a product rule to work:

scalar product

$$\large \frac{d}{dt}\left(f(t)\vec{r}(t)\right)=f'(t)\vec{r}(t)+f(t)\vec{r'}(t)$$

dot product

$$\large \frac{d}{dt}\left(\vec{r}(t)\cdot\vec{u}(t)\right)=\vec{r'}(t)\cdot\vec{u}(t)+\vec{r}(t)\cdot\vec{u'}(t)$$

cross product

$$\large \frac{d}{dt}\left(\vec{r}(t)\times\vec{u}(t)\right)=\vec{r'}(t)\times\vec{u}(t)+\vec{r}(t)\times\vec{u'}(t)$$

Integrals of Vector Valued Functions

As with derivatives, the integral of a vector valued function is the integral of each component:

$\large \int\vec{r}(t) dt=\left<\int x(t) dt,\int y(t) dt,\int z(t) dt \right>$

Just as in single variable calculus, we integral vector valued functions to undo derivatives. In particular, it is common to integrate to go from acceleration to velocity or velocity to position.

In the following video, we will see an example of a differentiation and integration for a given vector valued function:

Tomorrow's class:

Practice with derivatives and integrals!

Section Outcomes:

I can:

Find the derivative or integral of a vector valued function.
Find the tangent vector (or unit tangent vector) to a vector valued function at a point, or find a line tangent to a curve in space.
Find antiderivatives in motion problems.