Theory

Curves and line integrals

Parametric curves in the plane and in space, regularity and tangent vector, length and arc length, line integrals of the first and second kind with applications to mass, center of mass and work.

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Parametric curves
Regular curves and tangent vector

Parametric curves

1. Parametric curve and trace

A parametric curve is a continuous function $γ : [a, b] \to R^{2}$ (plane curve) or $γ : [a, b] \to R^{3}$ (space curve). In components one writes $γ (t) = (x (t), y (t))$ or $γ (t) = (x (t), y (t), z (t))$ ; the variable $t$ is called the parameter. The trace of $γ$ is the image $γ ([a, b])$ , that is, the set of points traversed. The curve is the function, not the set of points: the parametrization also contains the direction and the law by which the trace is traversed, and for this reason two different curves can have the same trace. All the integrals in this chapter are computed starting from a parametrization.

Same trace, different parametrizations

The curves

γ (t) = (cos t, sin t), t \in [0, 2 π], δ (t) = (cos 2 t, sin 2 t), t \in [0, π],

have the same trace, the unit circle, traversed counterclockwise starting from the point $(1, 0)$ . The second one traverses it in half the time, hence with twice the speed.

The unit circle: at the point

(1, 0)

the velocity vector of

δ

is twice that of

γ

2. Simple curves and closed curves

A curve $γ : [a, b] \to R^{n}$ is closed if $γ (a) = γ (b)$ , that is, if the final point coincides with the initial one. It is simple if it does not pass twice through the same point, that is, if $γ$ is injective on $[a, b]$ ; for a closed curve injectivity is required on $[a, b)$ , since the coincidence of the endpoints is part of the definition. The circle $(cos t, sin t)$ on $[0, 2 π]$ is closed and simple. Simple closed curves bound regions of the plane and appear in the chapter on Green’s theorem.

A simple closed curve, a curve that is not closed and a closed curve that passes twice through the same point

Remark

The graph of a function $f : [a, b] \to R$ is the trace of the curve $γ (t) = (t, f (t))$ , called a Cartesian curve. Every result on parametric curves therefore also applies to graphs; for Cartesian curves the formulas of the chapter take a simpler form, because $x^{'} (t) = 1$ .

Regular curves and tangent vector

1. Regular curve

A curve $γ : [a, b] \to R^{n}$ is regular if it is of class $C^{1}$ and $γ^{'} (t) \neq = 0$ for every $t \in (a, b)$ , where

γ^{'} (t) = (x^{'} (t), y^{'} (t)) or γ^{'} (t) = (x^{'} (t), y^{'} (t), z^{'} (t)) .

If $γ (t)$ is the position of a point at time $t$ , the vector $γ^{'} (t)$ is its velocity. The condition $γ^{'} (t) \neq = 0$ guarantees that at every point the curve has a well-defined direction of motion, hence a tangent line; it is this condition that makes the construction of line integrals possible.

2. Tangent vector and tangent line

Let $γ$ be regular and let $t_{0} \in (a, b)$ . The vector $γ^{'} (t_{0})$ is called the tangent vector to $γ$ at $γ (t_{0})$ ; the unit tangent vector is

T (t_{0}) = \frac{γ ^{'} ( t _{0} )}{∥ γ ^{'} ( t _{0} )∥} .

The tangent line to the curve at the point $γ (t_{0})$ is the line through $γ (t_{0})$ with direction $γ^{'} (t_{0})$ , with parametric equation

r (s) = γ (t_{0}) + s γ^{'} (t_{0}), s \in R .

The unit tangent vector indicates the direction and the orientation of travel; it appears in the relation between the two kinds of line integrals.

The ellipse

γ (t) = (2 cos t, sin t)

with the tangent vector and the tangent line at the point

γ (t_{0})

, for

t_{0} = \frac{π}{4}

The circular helix

The curve $γ (t) = (cos t, sin t, t)$ , $t \in [0, 2 π]$ , is the circular helix: its projection onto the $x y$ -plane traverses the unit circle while the height increases linearly. Since

γ^{'} (t) = (- sin t, cos t, 1), ∥ γ^{'} (t)∥ = sin^{2} t + cos^{2} t + 1 = 2,

the tangent vector is never zero, hence the curve is regular. The constant norm of the velocity vector makes the helix a recurring example in exercises on lengths and line integrals.

3. Piecewise regular curve

A continuous curve $γ : [a, b] \to R^{n}$ is piecewise regular if there exists a partition $a = t_{0} < t_{1} < \dots < t_{k} = b$ such that the restriction of $γ$ to each interval $[t_{i - 1}, t_{i}]$ is regular. Polygonal paths, formed by consecutive segments, and concatenations of arcs of regular curves are piecewise regular. Length and line integrals of a piecewise regular curve are computed by adding the contributions of the individual pieces; at the junction points the curve may have a corner.

A cusp

The curve $γ (t) = (t^{2}, t^{3})$ , $t \in [- 1, 1]$ , has $γ^{'} (t) = (2 t, 3 t^{2})$ , which vanishes for $t = 0$ . The curve is not regular on $[- 1, 1]$ , but it is piecewise regular, with the partition into the two intervals $[- 1, 0]$ and $[0, 1]$ . The trace has a cusp at the origin: the curve reaches the origin along the branch with $y < 0$ moving to the left, since $x^{'} (t) = 2 t < 0$ for $t < 0$ , and leaves along the branch with $y > 0$ moving to the right, with a reversal of the horizontal direction of motion.

The curve

γ (t) = (t^{2}, t^{3})

: the velocity vanishes for

t = 0

and the trace has a cusp at the origin

Warning

The vanishing of $γ^{'}$ at a point does not imply that the trace has a singular point there. The curve $γ (t) = (t^{3}, t^{3})$ , $t \in [- 1, 1]$ , has $γ^{'} (0) = (0, 0)$ , but its trace is the segment of the line $y = x$ between $(- 1, - 1)$ and $(1, 1)$ , which has no corners: the same segment is the trace of the regular curve $δ (t) = (t, t)$ . Regularity is a property of the parametrization, not of the trace.

The theory continues with 6 more sections (equivalent curves and orientation, length of a curve, arc length…), plus all worked examples, the solutions and the PDF.

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