Vector fields: Examples

Vector fields arise very naturally in physics and engineering applications from physical forces: gravitational, electrostatic, centrifugal, etc. For example, the vector field defined by the function
\[
\mathbf{F}(x,y,z)=-w_0\left(\frac{x}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}}\right),
\]
where $w_0$ is a real number, is associated with gravity and electrostatic attraction. The gravitational field around a planet and the electric field around a single point charge are similar to this field. The field points towards the origin (when $w_0>0$) and is inversely proportional to the square of the distance from the origin.


Simulation

Gravitational/Electrostatic field: Click on the image (or link below) to run the simulation.

Link: Here

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Another important example is the velocity vector field $\mathbf{v}$ of a steady-state fluid flow. The vector $\mathbf{v}(x, y)$ measures the instantaneous velocity of the fluid particles (molecules or atoms) as they pass through the point $(x, y)$. Steady-state means that the velocity at a point $(x, y)$ does not vary in time -even though the individual fluid particles are in motion. If a fluid particle moves along the curve $\mathbf{x}(t) = (x(t), y(t))$, then its velocity at time $t$ is the derivative
\[
\mathbf{v}= \frac{d\mathbf{x}}{dt}
\]
of its position with respect to $t$. Thus, for a time-independent velocity vector field
\[
\mathbf{v}(x, y) = ( v_1(x, y), v_2(x, y) )
\]
the fluid particles will move in accordance with an autonomous, first order system of ordinary differential equations
\[
\frac{dx}{dt}= v_1(x, y),\qquad \frac{dy}{dt}= v_2(x, y)
\]

According to the basic theory of systems of ordinary differential equations, an individual particle's motion $\mathbf{x}(t)$ will be uniquely determined solely by its initial position $\mathbf{x}(0) = \mathbf{x}_0$. In fluid mechanics, the trajectories of particles are known as the streamlines of the flow. The velocity vector $\mathbf{v}$ is everywhere tangent to the streamlines. When the flow is steady, the streamlines do not change in time. Individual fluid particles experience the same motion as they successively pass through a given point in the domain occupied by the fluid.

Examples of velocity vector fields of steady-state fluids flow are the following:

1. Rigid body rotation: $$\mathbf{v}(x,y)=(-wy,wx),\quad w\in \mathbb R.$$
2. Stagnation point: $$\mathbf{v}(x,y)=(kx,-ky), \quad k\in \mathbb R.$$
3. Vortex: $$\mathbf{v}(x,y)=\left(-\dfrac{y}{x^2+y^2},\dfrac{x}{x^2+y^2}\right).$$
4. Source and Sink: $$\mathbf{v}(x,y)=\dfrac{q}{2\pi}\left(\dfrac{x}{x^2+y^2},\dfrac{y}{x^2+y^2}\right),\quad q\in \mathbb R.$$

Simulation

The  following simulations show the steady-state fluid flows defined by the above velocity fields. Click on the image (or link below) to access the simulations.

Rigid body rotation

Link: Here
Stagnation point

Link: Here
Vortex

Link: Here
Source and Sink

Link: Here