VectorValued Functions
17 Calculus of VectorValued Functions
Learning Objectives
 Write an expression for the derivative of a vectorvalued 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 vectorvalued function.
To study the calculus of vectorvalued functions, we follow a similar path to the one we took in studying realvalued 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 VectorValued Functions
Now that we have seen what a vectorvalued function is and how to take its limit, the next step is to learn how to differentiate a vectorvalued function. The definition of the derivative of a vectorvalued function is nearly identical to the definition of a realvalued function of one variable. However, because the range of a vectorvalued function consists of vectors, the same is true for the range of the derivative of a vectorvalued function.
The derivative of a vectorvalued 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 realvalued functions can be applied to calculating the derivatives of vectorvalued functions as well. Recall that the derivative of a realvalued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function. The derivative of a vectorvalued 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 vectorvalued function.
Use the definition to calculate the derivative of the function
Let’s 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 vectorvalued function. This is always true for calculating the derivative of a vectorvalued 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 vectorvalued function and the derivative of a vectorvalued function.
Let f, g, and h be differentiable functions of t.
 If then
 If then
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.
 The first component of is The second component is We have and so the theorem gives
 The first component is and the second component is We have and so we obtain
 The first component of is the second component is and the third component is We have and so the theorem gives
We can extend to vectorvalued 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 vectorvalued functions. However, in the case of the product rule, there are actually three extensions: (1) for a realvalued function multiplied by a vectorvalued function, (2) for the dot product of two vectorvalued functions, and (3) for the cross product of two vectorvalued functions.
Let r and u be differentiable vectorvalued functions of t, let f be a differentiable realvalued 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 vectorvalued 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:
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Now for some examples using these properties.
Given the vectorvalued functions
and
calculate each of the following derivatives using the properties of the derivative of vectorvalued functions.
 We have and Therefore, according to property iv.:
 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,
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 vectorvalued function, the derivative provides a tangent vector to the curve represented by the function. Consider the vectorvalued 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
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
Let C be a curve defined by a vectorvalued 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.
Find the unit tangent vector for each of the following vectorvalued functions:
Find the unit tangent vector for the vectorvalued function
Follow the same steps as in (Figure).
Integrals of VectorValued Functions
We introduced antiderivatives of realvalued functions in Antiderivatives and definite integrals of realvalued functions in The Definite Integral. Each of these concepts can be extended to vectorvalued functions. Also, just as we can calculate the derivative of a vectorvalued function by differentiating the component functions separately, we can calculate the antiderivative in the same manner. Furthermore, the Fundamental Theorem of Calculus applies to vectorvalued functions as well.
The antiderivative of a vectorvalued function appears in applications. For example, if a vectorvalued 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.
Let f, g, and h be integrable realvalued functions over the closed interval
 The indefinite integral of a vectorvalued function is
The definite integral of a vectorvalued function is
 The indefinite integral of a vectorvalued function is
The definite integral of the vectorvalued function is
Since the indefinite integral of a vectorvalued 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 twodimensional 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.
Calculate each of the following integrals:
 We use the first part of the definition of the integral of a space curve:
 First calculate
Next, substitute this back into the integral and integrate:
 Use the second part of the definition of the integral of a space curve:
Calculate the following integral:
Use the definition of the definite integral of a plane curve.
Key Concepts
 To calculate the derivative of a vectorvalued function, calculate the derivatives of the component functions, then put them back into a new vectorvalued function.
 Many of the properties of differentiation from the Introduction to Derivatives also apply to vectorvalued functions.
 The derivative of a vectorvalued function is also a tangent vector to the curve. The unit tangent vector is calculated by dividing the derivative of a vectorvalued function by its magnitude.
 The antiderivative of a vectorvalued function is found by finding the antiderivatives of the component functions, then putting them back together in a vectorvalued function.
 The definite integral of a vectorvalued function is found by finding the definite integrals of the component functions, then putting them back together in a vectorvalued function.
Key Equations
 Derivative of a vectorvalued function
 Principal unit tangent vector
 Indefinite integral of a vectorvalued function
 Definite integral of a vectorvalued function
Compute the derivatives of the vectorvalued 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
 Graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function.
 Find the velocity as t approaches but is not equal to (if it exists).

 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 vectorvalued 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 vectorvalued 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 vectorvalued 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 vectorvalued functions.
The graph is shown here:
Evaluate the following integrals:
where
Glossary
 definite integral of a vectorvalued function
 the vector obtained by calculating the definite integral of each of the component functions of a given vectorvalued function, then using the results as the components of the resulting function
 derivative of a vectorvalued function
 the derivative of a vectorvalued function is provided the limit exists
 indefinite integral of a vectorvalued function
 a vectorvalued function with a derivative that is equal to a given vectorvalued 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