Page 100 - Vector Analysis

Chapter 4
Vector Calculus

4.1 The Line Integrals

4.1.1 Curves

Definition 4.1. A subset C Ď Rn is called a curve if C is the image of an interval I Ď R
under the continuous map γ : I Ñ Rn (that is, C = γ(I)). The continuous map γ : I Ñ Rn
is called a parametrization of the curve. A curve C is called simple if it has an injective
parametrization; that is, there exists γ : I Ñ Rn such that γ(I) = C and γ(x) = γ(y)
implies that x = y. A curve C with parametrization γ : I Ñ Rn is called closed if I = [a, b]
for some closed interval [a, b] Ď R and γ(a) = γ(b). A simple closed curve C is a closed
curve with parametrization γ : [a, b] Ñ Rn such that γ is one-to-one on (a, b).

Example 4.2. A line segment joining two points P0, P1 P Rn is a curve. It can be parame-
terized by γ : [0, 1] Ñ Rn defined by γ(t) = tP1 + (1 ´ t)P0.

Example 4.3. A circle on the plane is a simple closed curve. In fact, a circle centered at
the (x0, y0) with radius r has the following parametrization: γ : [0, 2π] Ñ R2 defined by
γ(θ) = (x0 + r cos θ, y0 + r sin θ).

Example 4.4. Figure eight is the zero level set of F (x, y) = x4 ´ a2(x2 ´ y2) for some a ‰ 0.

It can also be parameterized by γ : [0, 4π] Ñ R2 defined by γ(θ) = ( θ, a )
a cos sin θ .
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Definition 4.5 (Length of Curves). The length of curve C Ď Rn parameterized by γ :

[a, b] Ñ Rn is defined as the number

! k ˇ )
ˇ b
sup ÿ ››γ(ti) γ(ti´1)››Rn ˇ N and

ℓ(C ) ” ´ k P a = t0 ă t1 ă ¨¨ ¨ ă tk = .

i=1

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