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
\[
\mathbf{F}(x,y)=w_0\left(\frac{x}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}}\right),
\]
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.

Another important example is the velocity vector field $\mathbf{v}$ of a steadystate 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 $(x, y)$. Steadystate 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 timeindependent 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 steadystate 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 steadystate fluid flows defined by the above velocity fields. Click on the image (or link below) to access the simulations.
\[
\mathbf{F}(x,y)=w_0\left(\frac{x}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}},\frac{z}{\left(x^2+y^2+z^2\right)^{3/2}}\right),
\]
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.

Another important example is the velocity vector field $\mathbf{v}$ of a steadystate 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 $(x, y)$. Steadystate 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 timeindependent 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 steadystate 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 steadystate fluid flows defined by the above velocity fields. Click on the image (or link below) to access the simulations.
Rigid body rotation

Stagnation point

Vortex

Source and Sink

