Curve:
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Examples: The circle MATH for $t \in [0, \; 2\pi]$ is a simple closed curve. The oriented boundary of a regular polygon is a simple closed curve. Whereas, any curve of shape of the number $8$ is a closed curve, but not a simple curve. The straight line segment MATH for $t \in [0, \; 1]$ where $z_{1} \neq z_{2}$ is a simple curve, but not a closed curve. The curve MATH for $t \in \QTR{Bbb}{R}$ is a simple curve, but not a closed curve.

Jordan Curve Theorem: The points on any simple close curve (Jordan curve) $\gamma$ are boundary points of two distinct domains, one of which is the interior of $\gamma$ and is bounded. The other, which is the exterior of $\gamma$ is unbounded.

Definition (Differentiable Curve): A curve $\gamma: \gamma(t)$ ( $a \le t \le b$) is said to be a differentiable curve if the derivative $\gamma^{\prime}(t)$ exist and continuous for $a \le t \le b$.

Examples: The circle MATH for $t \in [0, \; 2\pi]$, the straight line segment MATH for $t \in [0, \; 1]$ and the curve MATH for $t \in \QTR{Bbb}{R}$ are differentiable curves.

Note: If MATH ( $a \le t \le b$) is a differentiable curve then the length of the curve $\gamma$ from $\gamma(a)$ and $\gamma(b)$ is given by
Suppose $C$ has a different parametric representation. Then also the value of the length of curve $L$ is invariant.

 
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