Two-Dimensional Calculus (2011)
Chapter 1. Background
2. Plane curves
Let f(t) be a function of the single variable t. Then f(t) is said to be differentiable in an interval a ≤ t ≤ b the derivative f'(t) exists at each point in the interval. It is continuously differentiable if the derivative f'(t) is a continuous function of t.
Definition 2.1 A regular (plane) curve C is defined by a pair of continuously differentiable functions x(t), y(t) in an interval a ≤ t ≤ b, satisfying
The significance of condition (2.1) will be explained later. Example 2.4 below is an illustration of what may happen if it does not hold. (See also Example 2.5 and Exs. 2.14 and 2.15.)
For the present we have a number of remarks to make concerning Def. 2.1.2 The chief observation to be made is that a curve, by our definition, is not simply a set of points in the plane, but is always described in terms of an additional variable t, called the parameter. The reason for defining curves in this way is that it is important to distinguish, for example, between a circle described once in the clockwise direction and the same circle described once in the counterclockwise direction or twice in the clockwise direction. The easiest way to make this distinction is to consider these as three different curves, even though they correspond to the same set of points in the plane. Intuitively we should think of a curve in the plane as a set of points together with a parameter, which in essence describes how the curve can be traced with a pencil.
The set of points defined by the equation
where f(x) is continuously differentiable, can be described as above by simply using x as the parameter. If we set
then
and condition (2.1) is satisfied. Thus Eq. (2.3) defines a regular curve. Equation (2.2) is often referred to as the explicit form or nonparametric form of this curve, in contrast to Eq. (2.3), which describes the curve in parametric form.
An equation of the form
is frequently said to define a curve in implicit form. The fact is that “in general” the set of points that satisfy such an equation can be traced out in the manner indicated. We shall return to this subject in sect. 8, where we give a precise meaning to the phrase “in general.”
Example 2.1
Let
Here a, b, c, d are arbitrary numbers. As t goes from 0 to 1, the point (x, y) defined by Eq. (2.5) moves along a straight line from (a, b) to (c, d) (Fig. 2.1). The geometric significance of the parameter t becomes clearer if we rewrite Eq. (2.5) in the form
FIGURE 2.1 Straight-line segment from (a, b) to (c, d)
These equations assert the equality of the ratios (x – a)/(c – a) and (y – b)/(d – b), with t as their common value (Fig. 2.2). When c − a = 0 or d – b = 0 we do not have this interpretation. However, when c − a = 0, the first of Eqs. (2.5a) yields x − a = 0 for all t, so that we obtain a segment of the vertical line x = a. If, on the other hand, c − a ≠ 0, then substituting t = (x − a)/(c − a) from the first of Eqs. (2.5a) into the second, we may rewrite Eqs. (2.5a) in the form
where x lies between a and c. This represents the same line segment in the explicit form of Eq. (2.2).
FIGURE 2.2 Geometric interpretation of parametric equations for a straight-line segment
Example 2.2
Let
Here c, d, and r are constants. Equation (2.6) defines the circle of radius r about the point (c, d), described once in counterclockwise direction and starting and ending at (c + r, d) (Fig. 2.3). The parameter t in this case is the angle, indicated in Fig. 2.3, measured in radians. This time we cannot expect to represent the curve in the explicit form of Eq. (2.2), since all values of x between c − r and c + r correspond to two distinct values of y. However, from Eq. (2.6) we find that
which is in the form of Eq. (2.4). Furthermore, given any point (x, y) satisfying Eq. (2.6a), we can find a value of the parameter t so that (x, y) satisfies Eq. (2.6). Thus Eq. (2.6a) can be regarded as the implicit form of a curve (in our sense); namely, the circle defined by Eq. (2.6).
Example 2.3
Let
where υ0, α, and g are constants. These equations represent the position after time t of a projectile having initial speed υ0 and initial inclination α, where g is the constant of gravitational acceleration (Fig. 2.4). This example is typical of many problems arising in physics, where the parameter t represents time (rather than
FIGURE 2.3 Circle of radius r
any geometrical quantity), and the functions x(t) and y(t) describe the position of a moving point at any time t.
If the quantity υ0 cos α is positive, then we may solve the first of Eqs. (2.7) as t = x/(υ0 COS α), and, substituting into the second, we find
which is, in explicit form, the equation of a parabolic arc.
FIGURE 2.4 Arc of parabola
Example 2.4
Let
This curve may easily be written in explicit form:
At the origin, however, this curve has a cusp with a vertical tangent (Fig. 2.5). This irregularity in the curve occurs despite the fact that x and y are both perfectly smooth differentiable functions of t. We have x'(t) = 3t2, y'(t) = 2t, which are certainly continuous functions of t. However,
and this expression vanishes when t = 0. Thus, condition (2.1) fails to be satisfied at the origin, and Eq. (2.8) is not a “regular curve" in our sense.
FIGURE 2.5 Example of a nonregular curve
It is worth noting that condition (2.1) is equivalent to the statement that x'(t) and y'(t) do not vanish simultaneously. Rather than verifying condition (2.1) directly, it is often easier to examine each of the equations x'(t) = 0 and y'(t) = 0 separately, and then check for common zeros.
Before turning to general properties of curves, we reconsider the first three examples above, and see what form they take in vector notation.
Example 2.1a
Equations (2.5a) may be written as a single vector equation
where v0 is a fixed vector:
Equation (2.9) expresses the fact that the displacement vector from (a, b) to (x, y) is a multiple of the fixed vector v0 (Fig. 2.6).
FIGURE 2.6 Straight-line segment in vector form
Example 2.2a
Equations (2.6) may be written in the form
This states that the displacement vector from (c, d) to (x, y) is a constant r times the unit vector COS t, sin t and hence has constant magnitude r (Fig. 2.7).
Example 2.3a
Equations (2,7) may be written as
where v0 and w are the fixed vectors
The vector v0 is known as the initial velocity vector. Equation (2.11) represents the displacement vector from the origin to the point (x, y) as the sum of two vectors. The first of these has the direction of the initial velocity vector and a magnitude that increases uniformly with t; the second is vertically downward in direction, and its magnitude grows at an increasing rate (Fig. 2.8).
FIGURE 2.7 Circle of radius r in vector form
Remark It is worth noting that the device used in Eq. (2.11) to write the curve (2.7) in vector form may be applied to an arbitrary curve. The position vector or radius vector of a point (x, y) is the vector x, y , which may be thought of as the displacement vector from the origin to the point (x, y). To each curve x(t), y(t) corresponds the vector function x(t), y(t) , which assigns to each value of t the position vector of the point (x(t), y(t)).
FIGURE 2.8 Arc of parabola in vector form
There are some formal advantages to this notation, but there is also a certain amount of confusion, which may result from identifying the point (x, y) with the vector x, y . We prefer to reserve vector notation for quantities that are fundamentally vectorial.
With the above examples in mind, we turn to a more general discussion of plane curves.
Definition 2.2 The length of the regular curve
is the number
It can be shown that this is the same value we obtain by a limit process if we approximate the curve by inscribed polygons.3 However, we simply use the integral in Eq. (2.13) as our definition of arc length.
If we consider only the part of the curve C corresponding to values of t between a and a fixed value t0, then the arc length of that part is equal to
Thus the arc length s is a function of the parameter value t0 (Fig. 2.9). By the fundamental theorem of calculus, the derivative of s is
Equation (2.15) expresses the rate of change of arc length with respect to the parameter t. When t represents time, the quantity s'(t0) is the speed at which the point is moving along the curve at the time t0.
With each point (x(t), y(t)) of the curve C we associate the vector
The condition (2.1) that the curve be regular, then takes the form
FIGURE 2.9 Arc-length function s(t0)
When t represents time, v(t) is called the velocity vector. By Eq. (2.15), the magnitude of the velocity vector is the speed
As for the direction of v(t), we note that by definition of the derivative,
so that4
The vectors on the right-hand side of Eq. (2.19) are in the direction of the displacement vector from (x(t0), y(t0)) to (x(t), y(t)). As t approaches t0, the direction of these vectors approaches the direction of the tangent to the curve (Fig. 2.10).
FIGURE 2.10 Velocity vector as a limit of displacement vectors
We may summarize as follows:
The vector v(t) defined by Eq. (2.16) at each point of the curve C has a direction tangent to the curve at the point, and magnitude equal to the derivative of arc length with respect to the parameter t.
The geometric interpretation of condition (2.1) is now clear. It implies that the vector v(t), being nonzero, has a well-defined direction, and hence the curve itself has a well-defined tangent direction at each point.
Definition 2.3 The vector T defined at each point of a regular curve C by
is called the unit tangent to the curve at the point.
It is often convenient to use the arc length s as a parameter for the curve. Specifically, to each value of s we may assign the coordinate (x, y) of the point at a distance s along the curve from its beginning. Then by the chain rule,
and using Eq. (2.18) we find that
We return to the examples given above, and see how these general considerations apply.
Example 2.1b
For the straight line of Eq. (2.5),
we have
Thus v(t) is a constant vector. If we denote its magnitude by m, so that
then by Eq. (2.18)
or
Substituting in the original equations of the line, we many express the coordinates explicity in terms of arc length:
Hence, using Eq. (2.20), we find the unit tangent T at each point is
Example 2.2b
For the circle
we find
and
Thus
and we may write
The unit tangent T is then (Fig. 2.11)
Note that it is customary to represent the velocity vector v = x'(t), y'(t) and the corresponding unit tangent T as displacement vectors starting at (x(t), y(t)).
FIGURE 2.11 Tangent vector to circle
Example 2.3b
For the parabola
we have
Hence
This example is much more typical of the general case in which we cannot hope to represent x and y as explicit functions of the arc length s, as we were able to in the special cases of the straight line and the circle. We may, though, always express the unit tangent T in the form v/|v|, so that here
It should be noted that often the velocity vector v is of more interest than T. In this particular case, we may rewrite Eq. (2.21) in the form
where v0 is the initial velocity vector υ0 COS α, υ0 sin α , and we see that the velocity vector at an arbitrary time t is the sum of the (fixed) initial velocity vector and a uniformly increasing vector in the negative y direction.
We conclude this section with an illustration of some of the procedures that may be used for sketching a curve given in parametric form.5
Example 2.5
Sketch the curve
We find
Since cos t ≤ 1 for all t, with equality holding for integer multiples of 2π, it follows that
On the other hand
and
Thus x'(t) and y'(t) vanish simultaneously for t = 0, ± 2π, ± 4π, .... In any interval excluding these values, condition (2.1) is satisfied, and we have a regular curve. The vector v = x'(t), y'(t) always points to the right, since its first component is positive. The curve has a horizontal tangent at the points t = ± π, ± 3π, ..., since at these points y'(t) = 0 and x'(t) ≠ 0. But these are precisely the points at which cos t = – 1, and at these points y takes on its maximum value 2. On the other hand, y ≥ 0 for all t, and y = 0 when cos t = 1.
We can now form a clear qualitative picture of the entire curve. As t increases, the corresponding point (x(t), y(t)) on the curve moves to the right, since x'(t) ≥ 0, while the y coordinate varies periodically between 0 and 2. The curve may therefore be described as a series of arches resting on the x axis at the points x = 0, ± 2π, .... It lies below the horizontal line y = 2, and touches that line at the points x = ± π, ± 3π, ....
To obtain a more precise picture, we note that
This means that adding 2π to the parameter value shifts the corresponding point on the curve a distance 2π to the right. In other words, each arch of the curve is obtained by translating the previous arch a distance 2π to the right. We therefore get a complete description of the curve once we have plotted a single arch, corresponding, say, to the interval 0 ≤ t ≤ 2π. For 0 < t < π, y increases with t, while for π < t < 2π, y decreases. Furthermore, since
the right half of the arch is symmetric to the left half; it is sufficient to draw the part of the curve corresponding to 0 ≤ t ≤ π, and then reflect across the line x = π. In Fig. 2.12, we have chosen a number of values of t between 0 and 2π, plotted the corresponding points (x(t), y(t)) on the curve, and drawn the vectors x'(t), y'(t) at each of these points. It is now a simple matter to sketch the complete curve.
FIGURE 2.12 Cycloid
The only question remaining is the precise behavior at the points 0, ± 2π, .... Since the vector x'(t), y'(t) vanishes at these points, we do not know if the curve has a well-defined direction. However, we may refer directly to the definition of the slope of a curve. We form the limit, as t tends to zero, of the slope m(t) of the line joining the origin to an arbitrary point (x(t), y(t)). We have
As t tends to zero, both the numerator and denominator on the right tend to zero. In order to determine whether or not a limit exists, we make a double application of l’Hôpital’s rule,6 which states that if the expression obtained by differentiating the numerator and denominator separately tends to a limit, then the original expression tends to the same limit; that is,
Thus the curve has a vertical tangent at the origin. It follows that at each of the values t = 0, ± 2π, ..., where condition (2.1) fails, the curve has a cusp with vertical tangent.
The curve defined by Eqs. (2.22) is a cycloid. It is the locus of a point on the circumference of a circle of radius 1 as the circle is rolled along the x axis.
Exercises
2.1 Write parametric equations for the following straight-line segments:
a. from (1, 1) to (4, 3)
b. from (1, −2) to (−3, 0)
c. from (2, 3) to (−1, 3)
d. starting at (−1, 1) with tangent vector T = , and length 10
e. starting at the origin, having length 2 and meeting the line y = x + 1 at right angles
2.2 Sketch each of the following curves, and express y explicitly as a function of x by eliminating the parameter t. (Observe carefully the limitation imposed on each curve by the interval prescribed for the parameter t.)
a.
b.
c.
d.
e.
2.3 Show that the curve
lies on a parabola. Sketch.
2.4 Show that the curve
is an ellipse, by obtaining an implicit equation of the form of Eq. (2.4).
2.5 Show that the curve
lies on a hyperbola. (Note that cosh t and sinh t are the hyperbolic functions— hyperbolic cosine and sine, respectively,
2.6 What conditions on the constants a, b, c, d are needed to guarantee that the line segment Eq. (2.5) of Example 2.1 be a regular curve? What do these conditions mean geometrically?
2.7 Repeat Ex. 2.6 for the constants c, d, r in Eq. (2.6) of Example 2.2.
*2.8 Repeat Ex. 2.6 for the constants υ0, α in Eq. (2.7) of Example 2.3.
2.9 Find the velocity vector v at an arbitrary point of each of the curves in Ex. 2.2.
2.10 Find the unit tangent T at an arbitrary point of each of the curves in Ex. 2.2.
2.11 Find the length of each of the following curves.
a.
b.
c.
d.
e.
*2.12 The folium of Descartes is defined by the equations
where t takes on all real values.
a. Write an equation for this curve in the implicit form of Eq. (2.4).
b. Sketch the curve.
(Hint: Note that the parameter t is equal to y/x, which is the slope of the line from the origin to the point (x, y) on the curve; x and y are both positive for t positive, whereas for t negative, x and y have opposite signs, but x + y = 3t/(1 – t + t2) ≥ – 1. It may also be helpful to sketch separately the graphs of x and y as functions of t.)
*2.13 The strophoid is given by the equation
a. Write an equation for this curve in the implicit form of Eq. (2.4).
b. Sketch the curve.
(Hint: Note that t = y/x, as in Ex. 2.12. Moreover x(t) = x(−t), y(t) = −y(−t) and − 1 ≤ x(t) ≤ 1, for all parameter values t. Consider the cases 0 ≤ t ≤ 1 and 1 ≤ t ≤ ∞.)
Exercises 2.14 and 2.15 are designed to give further insight into the significance of condition (2.1) in the definition of a regular curve.
2.14 Let C be the curve x = t3, y = t6, — 1 t 1.
a. Are x and y continuously differentiable functions of t ?
b. For which values of t is condition (2.1) satisfied?
c. Find an explicit expression for the curve C by expressing y as a function of x, and sketch the curve.
(Note that this example shows that when the condition (2.1) fails to hold the curve may have an irregularity, such as a cusp or a corner, but that need not be the case. The irregularity may arise from a “poor choice of parameter.”)
*2.15 Let f(t) be the function defined by
Let C be the curve defined by x = f(t), y = t2, −1 ≤ t ≤ 1.
Answer parts a, b, and c of Ex. 2.14.
2.16 At each point (x(t), y(t)) of a curve C, the vector
is called the acceleration vector. (It is defined only when the second derivatives of the functions x(t) and y(t) exist.)
a. Find the acceleration vector at an arbitrary point of each of the three curves given as Examples 2.1, 2.2, and 2.3 in the text.
b. Note in Example 2.2 that even though the speed |v(t)| is constant, the acceleration vector is not zero. Explain why this is so.
*c. Newton's second law of motion states that a force F acting on a particle of mass m induces an acceleration a such that F = ma.
If a particle moves in the plane under the action of a force directed toward the origin, then the vector F at each point is a scalar multiple of the radius vector x, y . (This situation is referred to as a central force field.) Show that under these conditions,
and
*d. Show conversely, that if a particle moves in such a way that x(dy/dt) − y(dx/dt) is constant, then at each point (x(t, y(t)) other than the origin, there is a scalar λ such that
Thus, by Newton’s second law, the force inducing the motion is always directed toward the origin. (Hint: Use Ex. 1.9a.)
2.17 Curves of the form x = a cos pt, y = b sin qt, where a, b, p, q are arbitrary positive constants, are known as Lissajous figures. When p and q are equal they lie on an ellipse (see Ex. 2.4). When p and q are integers they are closed curves described infinitely often, since both x and y are periodic in t. If the ratio p/q is not equal to a ratio of integers, then the curve swept out never repeats itself, although it always remains in the rectangle |x| ≤ a, |y| ≤ b. Sketch the following Lissajous figures for 0 ≤ t ≤ 2π.
a. x = COS t, y = sin 2t
b. x = COS 2t, y = sin t
*c. x = COS 3t, y = sin 2t
(Note. There are a number of mechanical methods for drawing Lissajous figures, such as compound pendulums, which vibrate in transverse directions. There are also practical applications.7 The variety of figures obtainable by simply varying the ratio p/q is astonishing. Some examples are shown in Fig. 2.13.)
*2.18 Let a regular curve be defined by x = φ(t), y = ψ(t), a ≤ t ≤ b, where φ'(t) > 0 for a ≤ t ≤ b. Using results from the calculus of functions of one variable, show that
a. the function x = φ(t), a ≤ t ≤ b has an inverse t = h(x), A ≤ x ≤ B, and substituting this inverse function in y = ψ(t) gives a nonparametric form of the curve: y = f(x), A ≤ x ≤ B;
b.
(Note. For an arbitrary regular curve, condition (2.1) implies that at each point either x'(t) ≠ 0 or y'(t) ≠ 0 (or both). As a consequence, it can be shown that the parameter interval for t can be divided into a finite number of intervals on each of which at least one of the conditions x'(t) > 0, x'(t) < 0, y'(t) > 0, or y'(t) < 0 holds throughout. It follows that on each of these intervals the curve can be expressed in nonparametric form, either as y = f(x) or x = g(y). Equation (2.13) is then a consequence of part b above and the arc-length formula for nonparametric curves.)
FIGURE 2.13 Lissajous figures [R. T. Lagemann.]