Curve:
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Example: Let MATH for $t \in [0, \; \pi]$. Then, MATH for $t \in [0, \; \pi]$. Further, MATH for $t \in [0, \; \pi]$. The length of the curve $\gamma$ joining $\gamma(0)$ and $\gamma(\pi)$ is MATH. Definition (Smooth Curve or Regular Curve): A curve $\gamma: \gamma(t)$ ( $a \le t \le b$) is said to be a smooth curve or regular curve if (i) the derivative $\gamma^{\prime}(t)$ exist and continuous for $a \le t \le b$ and (ii) MATH for $a < t < b$.

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 smooth curves.

Definition (Contour or Piecewise Smooth Curve): A contour or piecewise smooth curve , is a curve consisting of a finite number of smooth curves joined end to end

Examples: Any oriented polygonal path, circular path are contours.

Definition (Simple Closed Contour): A contour $\gamma: \gamma(t)$ ( $a \le t \le b$) having only the initial and final values are same is called a simple closed contour .

Examples: Any oriented triangle or rectangle or circle are examples of simple closed contours.

Definition (Opposite Curve): Consider the curve $\gamma$ having parametrization MATH for $a \le t \le b$. The opposite curve , denoted by $-\gamma$, traces out the same set of points in the complex plane but in the reverse order, and it has the parametrization  

 
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