A piecewise smooth curve is a type of curve that is composed of a finite number of smooth segments, meaning each segment is continuously differentiable, but the entire curve may have points where it is not differentiable. This concept is important as it allows for the integration and analysis of curves that may have sharp corners or breaks, which are relevant in both vector fields and complex analysis. Understanding how these curves behave is crucial for evaluating line integrals and applying Cauchy's integral theorem.
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