Vector-Valued Functions

17 Calculus of Vector-Valued Functions

Learning Objectives

• Write an expression for the derivative of a vector-valued function.
• Find the tangent vector at a point for a given position vector.
• Find the unit tangent vector at a point for a given position vector and explain its significance.
• Calculate the definite integral of a vector-valued function.

To study the calculus of vector-valued functions, we follow a similar path to the one we took in studying real-valued functions. First, we define the derivative, then we examine applications of the derivative, then we move on to defining integrals. However, we will find some interesting new ideas along the way as a result of the vector nature of these functions and the properties of space curves.

Derivatives of Vector-Valued Functions

Now that we have seen what a vector-valued function is and how to take its limit, the next step is to learn how to differentiate a vector-valued function. The definition of the derivative of a vector-valued function is nearly identical to the definition of a real-valued function of one variable. However, because the range of a vector-valued function consists of vectors, the same is true for the range of the derivative of a vector-valued function.

Definition

The derivative of a vector-valued function is

provided the limit exists. If exists, then r is differentiable at t. If exists for all t in an open interval then r is differentiable over the interval For the function to be differentiable over the closed interval the following two limits must exist as well:

Many of the rules for calculating derivatives of real-valued functions can be applied to calculating the derivatives of vector-valued functions as well. Recall that the derivative of a real-valued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function. The derivative of a vector-valued function can be understood to be an instantaneous rate of change as well; for example, when the function represents the position of an object at a given point in time, the derivative represents its velocity at that same point in time.

We now demonstrate taking the derivative of a vector-valued function.

Finding the Derivative of a Vector-Valued Function

Use the definition to calculate the derivative of the function

Let’s use (Figure):

Use the definition to calculate the derivative of the function

Hint

Use (Figure).

Notice that in the calculations in (Figure), we could also obtain the answer by first calculating the derivative of each component function, then putting these derivatives back into the vector-valued function. This is always true for calculating the derivative of a vector-valued function, whether it is in two or three dimensions. We state this in the following theorem. The proof of this theorem follows directly from the definitions of the limit of a vector-valued function and the derivative of a vector-valued function.

Differentiation of Vector-Valued Functions

Let f, g, and h be differentiable functions of t.

1. If then
2. If then
Calculating the Derivative of Vector-Valued Functions

Use (Figure) to calculate the derivative of each of the following functions.

We use (Figure) and what we know about differentiating functions of one variable.

1. The first component of is The second component is We have and so the theorem gives
2. The first component is and the second component is We have and so we obtain
3. The first component of is the second component is and the third component is We have and so the theorem gives

Calculate the derivative of the function

Hint

Identify the component functions and use (Figure).

We can extend to vector-valued functions the properties of the derivative that we presented in the Introduction to Derivatives. In particular, the constant multiple rule, the sum and difference rules, the product rule, and the chain rule all extend to vector-valued functions. However, in the case of the product rule, there are actually three extensions: (1) for a real-valued function multiplied by a vector-valued function, (2) for the dot product of two vector-valued functions, and (3) for the cross product of two vector-valued functions.

Properties of the Derivative of Vector-Valued Functions

Let r and u be differentiable vector-valued functions of t, let f be a differentiable real-valued function of t, and let c be a scalar.

Proof

The proofs of the first two properties follow directly from the definition of the derivative of a vector-valued function. The third property can be derived from the first two properties, along with the product rule from the Introduction to Derivatives. Let Then

To prove property iv. let and Then

The proof of property v. is similar to that of property iv. Property vi. can be proved using the chain rule. Last, property vii. follows from property iv:

Now for some examples using these properties.

Using the Properties of Derivatives of Vector-Valued Functions

Given the vector-valued functions

and

calculate each of the following derivatives using the properties of the derivative of vector-valued functions.

1. We have and Therefore, according to property iv.:
2. First, we need to adapt property v. for this problem:

Recall that the cross product of any vector with itself is zero. Furthermore, represents the second derivative of

Therefore,

Given the vector-valued functions and calculate and

Hint

Follow the same steps as in (Figure).

Tangent Vectors and Unit Tangent Vectors

Recall from the Introduction to Derivatives that the derivative at a point can be interpreted as the slope of the tangent line to the graph at that point. In the case of a vector-valued function, the derivative provides a tangent vector to the curve represented by the function. Consider the vector-valued function The derivative of this function is If we substitute the value into both functions we get

The graph of this function appears in (Figure), along with the vectors and

The tangent line at a point is calculated from the derivative of the vector-valued function

Notice that the vector is tangent to the circle at the point corresponding to This is an example of a tangent vector to the plane curve defined by

Definition

Let C be a curve defined by a vector-valued function r, and assume that exists when A tangent vector v at is any vector such that, when the tail of the vector is placed at point on the graph, vector v is tangent to curve C. Vector is an example of a tangent vector at point Furthermore, assume that The principal unit tangent vector at t is defined to be

provided

The unit tangent vector is exactly what it sounds like: a unit vector that is tangent to the curve. To calculate a unit tangent vector, first find the derivative Second, calculate the magnitude of the derivative. The third step is to divide the derivative by its magnitude.

Finding a Unit Tangent Vector

Find the unit tangent vector for each of the following vector-valued functions:

Find the unit tangent vector for the vector-valued function

Hint

Follow the same steps as in (Figure).

Integrals of Vector-Valued Functions

We introduced antiderivatives of real-valued functions in Antiderivatives and definite integrals of real-valued functions in The Definite Integral. Each of these concepts can be extended to vector-valued functions. Also, just as we can calculate the derivative of a vector-valued function by differentiating the component functions separately, we can calculate the antiderivative in the same manner. Furthermore, the Fundamental Theorem of Calculus applies to vector-valued functions as well.

The antiderivative of a vector-valued function appears in applications. For example, if a vector-valued function represents the velocity of an object at time t, then its antiderivative represents position. Or, if the function represents the acceleration of the object at a given time, then the antiderivative represents its velocity.

Definition

Let f, g, and h be integrable real-valued functions over the closed interval

1. The indefinite integral of a vector-valued function is

The definite integral of a vector-valued function is

2. The indefinite integral of a vector-valued function is

The definite integral of the vector-valued function is

Since the indefinite integral of a vector-valued function involves indefinite integrals of the component functions, each of these component integrals contains an integration constant. They can all be different. For example, in the two-dimensional case, we can have

where F and G are antiderivatives of f and g, respectively. Then

where Therefore, the integration constant becomes a constant vector.

Integrating Vector-Valued Functions

Calculate each of the following integrals:

1. We use the first part of the definition of the integral of a space curve:
2. First calculate

Next, substitute this back into the integral and integrate:

3. Use the second part of the definition of the integral of a space curve:

Calculate the following integral:

Hint

Use the definition of the definite integral of a plane curve.

Key Concepts

• To calculate the derivative of a vector-valued function, calculate the derivatives of the component functions, then put them back into a new vector-valued function.
• Many of the properties of differentiation from the Introduction to Derivatives also apply to vector-valued functions.
• The derivative of a vector-valued function is also a tangent vector to the curve. The unit tangent vector is calculated by dividing the derivative of a vector-valued function by its magnitude.
• The antiderivative of a vector-valued function is found by finding the antiderivatives of the component functions, then putting them back together in a vector-valued function.
• The definite integral of a vector-valued function is found by finding the definite integrals of the component functions, then putting them back together in a vector-valued function.

Key Equations

• Derivative of a vector-valued function
• Principal unit tangent vector
• Indefinite integral of a vector-valued function
• Definite integral of a vector-valued function

Compute the derivatives of the vector-valued functions.

A sketch of the graph is shown here. Notice the varying periodic nature of the graph.

For the following problems, find a tangent vector at the indicated value of t.

Find the unit tangent vector for the following parameterized curves.

Two views of this curve are presented here:

Let and Here is the graph of the function:

Find the following.

Compute the first, second, and third derivatives of

Find

The acceleration function, initial velocity, and initial position of a particle are

Find

The position vector of a particle is

1. Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function.
2. Find the velocity as t approaches but is not equal to (if it exists).

1. Undefined or infinite

Find the velocity and the speed of a particle with the position function The speed of a particle is the magnitude of the velocity and is represented by

A particle moves on a circular path of radius b according to the function where is the angular velocity,

Find the velocity function and show that is always orthogonal to

To show orthogonality, note that

Show that the speed of the particle is proportional to the angular velocity.

Evaluate given

Find the antiderivative of that satisfies the initial condition

Evaluate

An object starts from rest at point and moves with an acceleration of where is measured in feet per second per second. Find the location of the object after sec.

Show that if the speed of a particle traveling along a curve represented by a vector-valued function is constant, then the velocity function is always perpendicular to the acceleration function.

The last statement implies that the velocity and acceleration are perpendicular or orthogonal.

Given and find

Given find the velocity and the speed at any time.

Find the velocity vector for the function

Find the equation of the tangent line to the curve at

Describe and sketch the curve represented by the vector-valued function

Locate the highest point on the curve and give the value of the function at this point.

at

The position vector for a particle is The graph is shown here:

Find the velocity vector at any time.

Find the speed of the particle at time sec.

Find the acceleration at time sec.

A particle travels along the path of a helix with the equation See the graph presented here:

Find the following:

Velocity of the particle at any time

Speed of the particle at any time

Acceleration of the particle at any time

Find the unit tangent vector for the helix.

A particle travels along the path of an ellipse with the equation Find the following:

Velocity of the particle

Speed of the particle at

Acceleration of the particle at

Given the vector-valued function (graph is shown here), find the following:

Velocity

Speed

Acceleration

Find the minimum speed of a particle traveling along the curve

2

Given and find the following:

Now, use the product rule for the derivative of the cross product of two vectors and show this result is the same as the answer for the preceding problem.

Find the unit tangent vector T(t) for the following vector-valued functions.

The graph is shown here:

Evaluate the following integrals:

where

Glossary

definite integral of a vector-valued function
the vector obtained by calculating the definite integral of each of the component functions of a given vector-valued function, then using the results as the components of the resulting function
derivative of a vector-valued function
the derivative of a vector-valued function is provided the limit exists
indefinite integral of a vector-valued function
a vector-valued function with a derivative that is equal to a given vector-valued function
principal unit tangent vector
a unit vector tangent to a curve C
tangent vector
to at any vector v such that, when the tail of the vector is placed at point on the graph, vector v is tangent to curve C