Section 3 Key Concepts
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Suppose a curve \(C\) is parameterized by \(\vec{r}(t)\colon {\mathbb R} \rightarrow {\mathbb R}^n\text{:}\)
\(C\) is said to be continuous if the component functions of \(\vec{r}(t)\) are all continuous. It is piecewise continuous if the domain can be broken up into subintervals over each of which \(\vec{r}(t)\) is continuous.
\(C\) is said to be differentiable if the component functions of \(\vec{r}(t)\) are all differentiable. It is piecewise differentiable if the domain can be broken up into subintervals over each of which \(\vec{r}(t)\) is differentiable.
\(C\) is said to be \(C^1\) if the component functions of \(\vec{r}(t)\) all have continuous partial derivatives. It is called piecewise \(C^1\) if the domain can be broken up into subintervals over each of which \(\vec{r}(t)\) is \(C^1\text{.}\)
Accordingly, to test whether a curve is continuous, differentiable or \(C^1\text{,}\) we test whether the component functions of \(\vec{r}(t)\) exhibit this property.
A curve \(C\) is simple if it never intersects itself. In terms of parameterizations, a curve parametrized by \(\vec{r}(t)\) is simple if and only if \(\vec{r}(a)=\vec{r}(b)\) only when \(a=b\text{.}\)
A curve \(C\) is called a closed curve if the end point meets the initial point. In terms of parameterizations, a curve parametrized by \(\vec{r}(t)\) with \(a\leq t\leq b\) is closed if and only if \(\vec{r}(a)=\vec{r}(b)\text{.}\)
A path \(C\) is called a simple-closed curve if it is closed and the only point of intersection is the endpoint and initial point.