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A path or curve is a continuous mapping γ : [𝑎, b] → ℝn. We call γ((𝑎) the initial point and γ(b) the final point. The image of the path, γ([𝑎, b]), is called the arc of γ. If γ([𝑎, b]) ⊂ Ω, we say that γ is a path in Ω.
   
Example 1:    ■
End of Example 1
Recall that the derivative of γ(t) is
\[ \gamma' (t) = \lim_{\varepsilon\to 0} \frac{\gamma (t+\varepsilon ) - \gamma (t)}{\varepsilon} \]
if the limit exists. We observe that if t ∈ (𝑎, b), then γ′(t) exists if and only if γ is a differentiable mapping at the point t. In particular, we may consider γ′ : [𝑎, b] → ℝn, and where the appropriate limits exist, repeat to find higher-order derivatives of γ. This prompts the following definitions.
A path γ : [𝑎, b] → ℝn is said to be a function of class ℭp on [𝑎, b] if its pth derivative γ(p)(t) exists for every t ∈ [𝑎, b] and the p-th derivative is continuous on [𝑎, b]. The mapping γ is said to be a piecewise ℭp function if there exists a partition 𝑎 = t₀ < t₁ ⋯ < tn such that γ is of class ℭp[tk-1, tk] on every interval [tk-1, tk], k = 1, 2, … n.
We need one more definition.
A path γ : [𝑎, b] → ℝn is said to be smooth if γ is a ℭ¹ function and γ′(t) ≠ 0 for every t ∈ [𝑎, b].

 

 

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