Flow Line

A flow line of a vector field is a parametric curve that always lies tangential to it. That is, at every point of the curve, its derivative is given by the vector field. In the context of fluid flow, a flow line is usually referred to as a streamline.

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Definition §

Flow Line

A parametric curve $\vec{c}(t)$ is a flow line (or streamline) of a vector field $\vec{F}$ iff:

$$\frac{\text{d}\vec{c}}{\text{d}t}={\vec{c}}^{{}^{\prime }}(t)=\vec{F}(\vec{c}(t))$$

That is, the rate of change (or derivative) is given by the vector field.

Examples §

1: Proving a flow line

Show that the path $\vec{c}$ given by:

$$\vec{c}(t)=(\sin (t),\cos (t),2t)$$

is a flow line of the vector field $\vec{F}(x,y,z)=(y,-x,2)$

Solution

This is relatively simple. We just need to find ${\vec{c}}^{{}^{\prime }}(t)$ and check if it is equal to $\vec{F}(\vec{c}(t))$:

$$\begin{array}{rl}\frac{\text{d}\vec{c}}{\text{d}t}& =(\cos (t),-\sin (t),2)\\ \vec{F}(\vec{c}(t))& =\vec{F}(\sin (t),\cos (t),2t)=(\cos (t),-\sin (t),2)\end{array}$$

Hence, ${\vec{c}}^{{}^{\prime }}(t)=\vec{F}(\vec{c}(t))$ and $\vec{c}$ is a flow line of $\vec{F}$.

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