4 Vector fields

A vector field is a distribution of vectors over a region of space such that a vector is associated with each point of the region. Examples are:

1. The velocity of water flowing in a river (Figure 7).

   Figure 7:

   

2. The gravitational pull of the Earth (Figure 8). At every point there is a gravitational pull towards the centre of the Earth.

   Figure 8:

   

   Note: the length of the vector is used to indicate its magnitude (i.e. greater near the centre of the Earth.)

3. The flow of heat in a metal plate insulated on its sides (Figure 9). Heat flows from the hot portion on the left to the cool portion on the right.

   Figure 9:

   

To define a vector field we need to :

• Describe the region of space where the vectors are found (the domain)
• Give a rule for associating a vector with each point of the domain.

Note that in the case of the heat flowing in a plate, the temperature can be described by a scalar field while the flow of heat is described by a vector field.

Consider the flow of water in different situations.

1. In a pond where the water is motionless everywhere, the velocity at all points is zero. That is, $\underset{̲}{v}\left(x,y,z\right)=\underset{̲}{0}$ , or for brevity, $\underset{̲}{v}=\underset{̲}{0}$ .
2. Consider a straight river with steady flow downstream (see Figure 10). The surface velocity $\underset{̲}{v}$ can be seen by watching the motion of a light floating object, such as a leaf. The leaf will float downstream parallel to the bank so $\underset{̲}{v}$ will be a multiple of $\underset{̲}{j}$ . However, the speed is usually smallest near the bank and fastest in the middle of the river. In this simple model, the velocity $\underset{̲}{v}$ is assumed to be independent of the depth $z$ . That is, $\underset{̲}{v}$ varies, in the $\underset{̲}{i}$ , or $x$ , direction so that $\underset{̲}{v}$ will be of the form $\underset{̲}{v}=f\left(x\right)\underset{̲}{j}$ .

   Figure 10:

   

3. In a more realistic model $\underset{̲}{v}$ would vary as we move downstream and would be different at different depths due to, for example, rocks or bends. The velocity at any point could also depend on when the observation was made (for example the speed would be higher shortly after heavy rain) and so in general the velocity would be a function of the four variables $x$ , $y$ , $z$ and $t$ , and be of the form $\underset{̲}{v}=f_1\left(x,y,z,t\right)\underset{̲}{i}+f_2\left(x,y,z,t\right)\underset{̲}{j}+f_3\left(x,y,z,t\right)\underset{̲}{k},$ for suitable functions $f_1,f_2$ and $f_3$ .

Example 7

Sketch sample vectors at the points $\left(3,2\right),\left(-2,2\right),\left(-3,-1\right),\left(-1,4\right)$ for the two-dimensional vector field defined by $\underset{̲}{v}=x\underset{̲}{i}+2\underset{̲}{j}$ .

Solution

 

At $\left(3,2\right)$ , $\underset{̲}{v}=3\underset{̲}{i}+2\underset{̲}{j}$

At $\left(-2,2\right)$ , $\underset{̲}{v}=-2\underset{̲}{i}+2\underset{̲}{j}$

At $\left(-3,-1\right)$ , $\underset{̲}{v}=-3\underset{̲}{i}+2\underset{̲}{j}$

At $\left(-1,4\right)$ , $\underset{̲}{v}=-\underset{̲}{i}+2\underset{̲}{j}$

Plotting these vectors $\underset{̲}{v}$ gives the arrows in Figure 11.

Figure 11:

It is possible to construct curves which start from and are in the same direction as any one vector and are guided by the direction of successive vectors. Starting at different points gives a set of non-intersecting lines called, depending on the context, vector field lines, lines of flow, streamlines or lines of force.

For example, consider the vector field $\underset{̲}{F}=-y\underset{̲}{i}+x\underset{̲}{j}$ ; $\underset{̲}{F}$ can be calculated at various points in the $xy$ plane. Some of the individual vectors can be seen in Figure 12(a) while Figure 12(b) shows them converted seamlessly to field lines. For this function $\underset{̲}{F}$ the field lines are circles centered at the origin.

Figure 12:

Example 8

The Earth is affected by the gravitational force field of the Sun. This vector field is such that each vector $\underset{̲}{F}$ is directed towards the Sun and has magnitude proportional to $\frac{1}{r^2}$ , where $r=\sqrt{x^2+y^2+z^2}$ is the distance from the Sun to the Earth. Derive an equation for $\underset{̲}{F}$ and sketch some field lines.

Solution

The field has magnitude proportional to $r^{-2}={\left(x^2+y^2+z^2\right)}^{-1}$ and points directly towards the Sun (the origin) i.e. parallel to a unit vector pointing towards the origin. At the point given by $\underset{̲}{r}=x\underset{̲}{i}+y\underset{̲}{j}+z\underset{̲}{k}$ , a unit vector pointing towards the origin is $\frac{-x\underset{̲}{i}-y\underset{̲}{j}-z\underset{̲}{k}}{\left|-x\underset{̲}{i}-y\underset{̲}{j}-z\underset{̲}{k}\right|}=\frac{-x\underset{̲}{i}-y\underset{̲}{j}-z\underset{̲}{k}}{\sqrt{x^2+y^2+z^2}}$ . Multiplying the unit vector by the required magnitude $r^{-2}={\left(x^2+y^2+z^2\right)}^{-1}$ (and by a constant of proportionality $c$ ) gives $\underset{̲}{F}=c\frac{-x\underset{̲}{i}-y\underset{̲}{j}-z\underset{̲}{k}}{{\left(x^2+y^2+z^2\right)}^{3∕2}}$ . Figure 13 shows some field lines for $\underset{̲}{F}$ .

Figure 13:

Exercises

1. Which of the following are scalar fields and which are vector fields?
   1. $F=x^2-yz$
   2. $G=\frac{2x-z}{\sqrt{x^2+y^2+z^2+1}}$
   3. $\underset{̲}{f}=x\underset{̲}{i}+y\underset{̲}{j}+z\underset{̲}{k}$
   4. $H=\frac{y-1}{z^2+1}x+\frac{z-1}{x^2+1}y+\frac{x-1}{y^2+1}z$
   5. $\underset{̲}{g}=\left(y+z\right)\underset{̲}{i}$
2. Draw streamlines for the vector fields
   1. $\underset{̲}{f}=\underset{̲}{i}+2\underset{̲}{j}$
   2. $\underset{̲}{g}=\underset{̲}{i}+y^2\underset{̲}{j}$

Answer