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MATH
Example: Let MATH for $0 \le t \le 2 \pi$. Then, the parametric representation of the opposite curve $-C$ is given by MATH for $-2 \pi \le t \le 0$.

Orientation of a curve: Let $\gamma$ be a curve with the parametrization $\gamma = \gamma(t)$ for $t \in [a, \; b]$. As $t$ increases from $a$ to $b$, the points $\gamma(t)$ moves continuously from $\gamma(a)$ to $\gamma(b)$ in a specific direction which we indicate by drawing arrows along the curve. This direction is called the orientation (or natural orientation) of the curve induced by the parametrization $\gamma = \gamma(t)$ for $t \in [a, \; b]$.

Positive/Negative Orientation: Let $\gamma$ be a simple closed contour with the parametrization MATH for $t \in [a, \; b]$. If $\gamma$ is parametrized so that the interior bounded domain of $\gamma$ is kept on the left as $z(t)$ moves around $\gamma$, then we say that $\gamma$ is oriented in the positive (counterclockwise or anticlockwise) sense; otherwise, $\gamma$ is oriented negatively (clockwise).

Examples: The circle MATH for $0 \le t \le 2 \pi$ is oriented positively. The circle MATH for MATH is oriented negatively.
Note: If a simple closed curve $\gamma$ is positively oriented, then the opposite curve $-\gamma$ is negatively oriented. If the orientation (or parametrization) of a simple closed curve $\gamma$ is not given, then it is understood that the simple closed curve $\gamma$ is oriented positively.

 
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