Basic Definitions

Section 7.1 Basic Definitions

In this section we aim to give a definition of “curves”, and then to discuss some possible properties. Suppose that \(\vect\gamma\) is a continuous vector valued function defined on an interval \(I\subset\mathbb R\) with values in \(\mathbb R^N\text{.}\) By definition, the image,

\begin{equation*} C:=\{\vect\gamma(t)\colon t\in I\}\subset\mathbb R^N\text{,} \end{equation*}

is called a curve in \(\mathbb R^N\text{.}\) As discussed already in Section 2.4, the function \(\vect\gamma\) has \(N\) component functions, namely

\begin{equation*} \vect\gamma(t)=\bigl(\gamma_1(t),\gamma_2(t),\dots,\gamma_N(t)\bigr) \end{equation*}

for \(t\in I\text{.}\) If \(N=2\) or \(N=3\) we can visualise the curve \(C\) as shown in Figure 2.14. We can think of \(C\) as the path followed by a particle moving in space. If \(C\) has no points of intersection with itself, then we call \(C\) a simple curve, if \(C\) has the same start and endpoints it is called a closed curve. See Figure 7.1 to visualise this.

Obviously there are many ways to travel along a given path. We can move slower or faster, backward and forward, etc. Hence there are many different continuous functions describing the same curve. We call every one-to-one function describing \(C\) a parametrisation of \(C\) .

Definition 7.2. Smooth curve.

We call \(C\) a smooth curve if it admits a continuously differentiable parametrisation \(\vect\gamma(t)\text{,}\) \(t\in I\text{,}\) such that \(\vect\gamma'(t)\neq\vect 0\) for all \(t\in I\text{.}\) A parametrisation as the above is called a regular parametrisation of \(C\text{.}\) We call \(C\) a piecewise smooth curve if it can be written as a union of finitely many smooth curves.

Figure 7.3 shows a piecewise smooth curve, the curves shown in Figure 7.1 are all smooth. In this course we will always work with smooth or piecewise smooth curves.

We saw in Section 4.2 that \(\vect\gamma'(t)\) points in the direction of the tangent of \(C\) at \(\vect\gamma(t)\text{.}\) If \(C\) is a smooth curve and \(\vect\gamma(t)\text{,}\) \(t\in I\text{,}\) is a regular parametrisation of \(C\) we can compute a unit tangent vector at every point of \(C\text{.}\) As \(\vect\gamma'(t)\neq\vect 0\) for all \(t\) we can set

\begin{equation} \vect\tau(t):=\frac{\vect\gamma'(t)}{\|\vect\gamma'(t)\|}, \qquad t\in I\text{.}\tag{7.1} \end{equation}

At each point of the curve there are exactly two unit tangent vectors pointing in opposite directions. If \(\vect\tau(t)\) is given as above then \(\vect\tau(t)\) and \(-\vect\tau(t)\) are the only continuous unit tangent vectors. One of them we define to be the positive field of unit tangents. The opposite field is called the negative field of unit tangents. The fields are depicted in Figure 7.4.

Figure 7.4. The two possible orientations of a curve.

The two possibilities reflect the fact that we can run along the curve in two different directions, one called positive, the opposite one called negative.

Definition 7.5. Orientation of a curve.

A smooth curve, \(C\text{,}\) with the positive unit tangent vector field is called positively oriented. Equipped with the opposite unit tangent vector field \(C\) is called negatively oriented. We often write

\(C\) for the positively oriented curve;
\(-C\) for the negatively oriented curve;
\(|C|\) for the curve without a particular orientation fixed.

We call a parametrisation, \(\vect\gamma(t)\text{,}\) \(t\in I\) consistent with the orientation of the curve if (7.1) defines the positive field of tangents.

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