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As the velocity of a moving fluid and the flow of an electric current which lead, in turn, to important physical and engineering applications. We begin our study of integration theory by defining the integral of a vector field F along a directed curve . Let P : [a, b] → denote a parametrization of . To each partition of [a, b] we obtain a partition of (Fig. 2) and the Riemann sum i F P(ti ) · P(ti+1 ) − P(ti ) ≈ i F P(ti ) · P ′ (ti ) ti where ti = ti+1 − ti and · denotes the inner product in Rn .

Since we allow A = B it follows that the mapping P may not be injective on [a, b]. However, we do not wish the curve to cross itself (Fig. 3a) or to half cross itself (Fig. 3b) as these lead to unnecessary complications and we have included condition (d) to exclude such possibilities. A continuous mapping P : [a, b] → Rn which satisfies (a), (c) and (d) is called a parametrized curve. A parametrized curve determines precisely one directed curve P([a, b]), P(a), P(b), P ′ (a) ∥P ′ (a)∥ for which it is a parametrization.

Since P is differentiable we have for all t and t + ∆t in [a, b] P(t + ∆t) = P(t) + P ′ (t)∆t + g(t, ∆t) · ∆t 5 Curves in Rn 50 P (b) P (ti+1 ) P (ti ) ∆ti a ti ti+1 b P (a) Fig. 4 where g(t, ∆t) → 0 as ∆t → 0 for any fixed t. Hence P(t + ∆t) − P(t) ≈ P ′ (t)∆t for ∆t close to zero. If we partition [a, b] we get a corresponding partition of Γ and an approximation of the length of Γ (Fig. 4). We have l(Γ ) ≈ i ≈ i ∥P(ti+1 ) − P(ti )∥ ∥P ′ (ti )∥∆ti b −−−→ a ∥P ′ (t)∥dt as we take finer and finer partitions of [a, b].