11.2 The Calculus of Vector-Valued Functions

Now that we have defined vector-valued functions, we need some tools for examining them. In this section, we begin to explore the calculus of vector-valued functions. As with scalar functions, we begin with the notion of limit and progress to continuity, derivatives and finally, integrals. Take careful note of how our presentation parallels that from Chapters 1, 2 and 4. We follow this same kind of progression again when we examine functions of several variables in Chapter 12. We define everything in this section in terms of vector-valued functions in three dimensions. The definitions can be interpreted for vector-valued functions in two dimensions in the obvious way, by simply dropping the third component everywhere.

For a vector-valued function , if we write

we mean that as t  gets closer and closer to a, the vector r(t ) is getting closer and closer to the vector u . If we write , this means that
Notice that for this to occur, we must have that f (t ) is approaching u1, g(t ) is approaching u2 and h(t ) is approaching u3. In view of this, we make the following definition.

2.1   
 
For a vector-valued function the limit of r(t ) as t  approaches a is given by
Limit of a vector-valued function (2.1)

provided all of the indicated limits exist. If any of the limits indicated on the right-hand side of (2.1) fail to exist, then does not exist.
 

In the following example, we see that calculating a limit of a vector-valued function simply consists of calculating three separate limits of scalar functions.

2.1   
Finding the Limit of a Vector-Valued Function
 
Find
 
 
Recall that each of the component functions is continuous (for all t  ) and so, we can calculate their limits simply by substituting the values for t  . We have
 

2.2   
A Limit That Does Not Exist
 
Find
 
 
Notice that the limit of the third component is which does not exist. So, even though the limits of the first two components exist, the limit of the vector-valued function does not exist.
 

Recall that for a scalar function f  , we say that f  is continuous at a if and only if
That is, a scalar function is continuous at a point whenever the limit and the value of the function are the same. We define the continuity of vector-valued functions in the same way.

2.2   
 
The vector-valued function is continuous at t = a whenever
 Continuity of a vector-valued function
(i.e., whenever the limit and the value of the vector-valued function are the same).
 

Notice that in terms of the components of r, this says that r(t ) is continuous at t = a whenever
Further, since
it follows that r is continuous at t = a if and only if
Finally, note that this occurs if and only if
If you look carefully at what we have just said, youšll notice that we just proved the following theorem:

2.1   
 
A vector-valued function is continuous at t = a if and only if all of f, g and h are continuous at t = a.
 

Notice that Theorem 2.1 says that if you want to determine whether or not a vector-valued function is continuous, you need only check the continuity of each component function (something you already know how to do!). We demonstrate this in the following two examples.

2.3   
Determining Where a Vector-Valued Function is Continuous
 
Determine for what values of t  the vector-valued function is continuous.
 
 
From Theorem 2.1, we need only consider the continuity of the component functions; r(t ) will be continuous wherever all its components are continuous. We have: e5t  is continuous for all t, ln (t+1) is continuous for t > -1 and cos t  is continuous for all t. This says that r(t ) is continuous for t > -1.
 

2.4   
A Vector-Valued Function with Infinitely Many Discontinuities
 
Determine for what values of t  the vector-valued function is continuous.
 
 
First, note that tan t  is continuous, except at for n = 0, 1, 2, (i.e., tan t  is continuous except at odd multiples of ). The second com-ponent | t+3| is continuous for all t  (although it's not differentiable at t = -3 ). Finally, the third component is continuous except at t = 2. Since all three components must be continuous in order for r(t ) to be continuous, we have that r(t ) is continuous, except at t = 2 and t = for n = 0, 1, 2, .
 

Recall that in Chapter 2, we defined the derivative of a scalar function f  to be
Replacing h by t, we can rewrite this as
You may be wondering why we want to change from a perfectly nice variable like h to something more unusual like t. The only reason is that we want to use the notation to emphasize that t  is an increment of the variable t. In Chapter 12, we'll be defining partial derivatives of functions of more than one variable, where we'll use this type of notation to make it clear which variable is being incremented.

We now define the derivative of a vector-valued function in the expected way.

2.3   
 
The derivative r' (t ) of the vector-valued function r(t ) is defined by
(2.2)
for any values of t  for which the limit exists. When the limit exists for t = a, we say that r is differentiable at t = a.
 



Fortunately, you will not need to learn any new differentiation rules, as the derivative of a vector-valued function is found directly from the derivatives of the individual components, as we see in the following result.

2.2   
 
Let and suppose that the components f, g and h are all differentiable for some value of t. Then r is also differentiable at that value of t  and its derivative is given by
(2.3)
   
 
From the definition of derivative of a vector-valued function (2.2), we have
 
 

where we have used the definition of vector subtraction. Distributing the scalar into each component and using the definition of limit of a vector-valued function (2.1), we have

 
 
 

where in the last step we recognized the definition of the derivatives of each of the component functions f, g and h.
 

Notice that thanks to Theorem 2.2, in order to differentiate a vector-valued function, we need only differentiate the individual component functions, using the usual rules of differentiation. We illustrate this in the following example.

2.5   
Finding the Derivative of a Vector-Valued Function
 
Find the derivative of
 
 
Applying the chain rule to the first two components and the product rule to the third, we have (for t > 0 ):
 
 
 

For the most part, to compute derivatives of vector-valued functions, we only need to use the already familiar rules for differentiation of scalar functions. There are several special derivative rules, however, which we state in the following theorem.

2.3   
 
Suppose that r(t ) and s(t ) are differentiable vector-valued functions, f (t ) is a differentiable scalar function and c is any scalar constant. Then
(i)
(ii)
(iii)
(iv)
(v)

Notice that parts (iii), (iv) and (v) are the product rules for the various kinds of products we can define. In (iii), we have the derivative of a product of a scalar function and a vector-valued function; in (iv) we have the derivative of a dot product and in (v), we have the derivative of a cross product. In each of these three cases, it's important for you to recognize that these follow the same pattern as the product rule for the derivative of the product of two scalar functions.

   
 
(i) For and we have from (2.3) and the rules for vector addition that
 
 
 
  = r' (t )+s' (t ).
(iv) From the definition of dot product and the usual product rule for the product of two scalar functions, we have
 
  = f 1' (t )f 2(t )+f 1(t )f 2' (t )+g1' (t )g2(t )+g1(t )g2'(t )
      + h1' (t )h2(t )+h1(t )h2' (t )
  = [f 1' (t )f 2(t )+g1' (t )g2(t )+h1' (t )h2(t )]
      + [ f 1(t )f 2' (t )+g1(t )g2' (t )+h1(t )h2' (t )]
  = r' (t ) s(t )+r(t ) s ' (t ).
We leave the proofs of (ii), (iii) and (v) as exercises.
 

We next explore an important graphical interpretation of the derivative of a vector-valued function. First, recall that one interpretation of the derivative of a scalar function is that the value of the derivative at a point gives the slope of the tangent line to the curve at that point. For the case of the vector-valued function r(t ), notice that from (2.2), the derivative of r(t ) at t = a is given by
Again, recall that the endpoint of the vector-valued function r(t ) traces out a curve C in 3. In Figure 11.9a, we show the position vectors r(a) , r(a+ t ) and r (a+ t )- r(a), for some fixed t > 0, using our graphical interpretation of vector subtraction, developed in Chapter 10. (How does the picture differ if t < 0 ?) Notice that for t > 0, the vector points in the same direction as r (a+ t )- r(a). If we take smaller and smaller values of t, will approach r' (a). We illustrate this graphically in Figures 11.9b and 11.9c. As t 0, notice that the vector approaches a vector that is tangent to the curve C at the terminal point of r (a), as seen in Figure 11.9d. We refer to



Figure 11.9a



Figure 11.9b



Figure 11.9c



Figure 11.9d
The tangent vector r' (a).

r' (a) as the tangent vector to the curve C at the point corresponding to t = a. Be sure to observe that r' (a) lies along the tangent line to the curve at t = a and points in the direction of the orientation of C. (Recognize that Figures 11.9a, 11.9b and 11.9c are all drawn so that t > 0. What changes in each of the figures if t < 0? )

We illustrate this notion for a simple curve in 2 in the following example.

2.6   
Drawing Position and Tangent Vectors
 
For , plot the curve traced out by the endpoint of r (t ) and draw the position vector and tangent vector at
 
 
First, notice that
Also, the curve traced out by r(t ) is given parametrically by
You should observe that, in this case,
x2+y2 = cos 22t+sin 22t = 1,
so that the curve is the circle of radius 1, centered at the origin. Further, from the parameterization, you can see that the orientation is clockwise. The position and tangent vectors at are given by
and
respectively. We show the curve, along with the vectors and in Figure 11.10. In particular, you might note that
so that and are orthogonal. In fact, r(t ) and r' (t ) are orthogonal for every t, as follows:
r(t ) r'(t ) = -cos2t, sin2t2 sin2t, 2cos2t
  = -2cos 2t sin 2t+2 sin 2t cos 2t = 0.



Figure 11.10
Position and tangent vectors.
 

Were you surprised to find in example 2.6 that the position vector and the tangent vector were orthogonal at every point? As it turns out, this is a special case of a more general result, which we state in the following theorem.

2.4   
 
constant if and only if r(t ) and r' (t ) are orthogonal, for all t.
 
   
 
(i) Suppose that for some constant c. Recall that
(2.4)

Differentiating both sides of (2.4), we get

From Theorem 2.3 (iv), we now have

so that r(t ) r' (t ) = 0, as desired.

(ii) We leave the proof of the converse as an exercise.
 

You should observe that Theorem 2.4 has some geometric significance. First, note that in two dimensions, if (where c is a constant), then the curve traced out by the position vector r (t ) must lie on the circle of radius c , centered at the origin. (Think about this some!) We can then interpret Theorem 2.4 to say that the path traced out by r(t ) lies on a circle centered at the origin if and only if the tangent vector is orthogonal to the position vector at every point on the curve. Likewise, in three dimensions, if (where c is a constant), the curve traced out by r(t ) lies on the sphere of radius c centered at the origin. In this case, we can interpret Theorem 2.4 to say that the curve traced out by r(t ) lies on a sphere centered at the origin if and only if the tangent vector is orthogonal to the position vector at every point on the curve.

We conclude this section by making a few straightforward definitions. Recall that when we say that the scalar function F(t ) is an antiderivative of the scalar function f (t ), we mean that F is any function such that F' (t ) = f (t ). We extend this notion to vector-valued functions in the following definition.

2.4   
 
The vector-valued function R(t ) is an antiderivative of the vector-valued function r(t ) whenever R' (t ) = r(t ).

Notice that if and f, g and h have antiderivatives F, G and H, respectively, then
That is, is an antiderivative of r(t ). In fact, is also an antiderivative of r(t ), for any choice of constants c1, c2 and c3. This leads us to the following definition.

2.5   
 
If R(t ) is any antiderivative of r(t ), the indefinite integral of r(t ) is defined to be
where c is an arbitrary constant vector.
 

As in the scalar case, R(t )+ c is the most general antiderivative of r(t ) . (Why is that?) Notice that this says that
Indefinite integral of a vector-valued function (2.5)
That is, you integrate a vector-valued function by integrating each of the individual components.

2.7   
Evaluating the Indefinite Integral of a Vector-Valued Function
 
Evaluate the indefinite integral
 
 
From (2.5), we have
where is an arbitrary constant vector.
 

Similarly, we define the definite integral of a vector-valued function in the obvious way.

2.6   
 
For the vector-valued function , we define the definite integral of r(t ) by
Definite integral of a vector-valued function (2.6)

Notice that this says that the definite integral of a vector-valued function r(t ) is simply the vector whose components are the definite integrals of the corresponding components of r(t ) . With this in mind, we now extend the Fundamental Theorem of Calculus to vector-valued functions.

2.5   
 
Suppose that R(t ) is an antiderivative of r(t ) on the interval [ a, b]. Then,

   
 
The proof is straightforward and we leave this as an exercise.
 

2.8   
Evaluating the Definite Integral of a Vector-Valued Function
 
Evaluate
 
 
Notice that an antiderivative for the integrand is
From Theorem 2.5, we have that
 
 

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