16.1 Vector Fields

16.1 Vector Fields

Vector Fields

The vectors in Figure 1 are air velocity vectors that indicate the wind speed and direction at points 10 m above the

surface elevation in the San Francisco Bay area.

We see at a glance from the largest arrows in part (a) that the greatest wind speeds at that time occurred as the winds entered the bay across the Golden Gate Bridge. Part (b)

shows the very different wind pattern 12 hours earlier.

Vector Fields

Vector Fields

Associated with every point in the air we can imagine a

wind velocity vector. This is an example of a velocity vector field.

Other examples of velocity vector fields are illustrated in Figure 2: ocean currents and flow past an airfoil.

Velocity vector fields

Vector Fields

Another type of vector field, called a force field, associates a force vector with each point in a region. An example is the gravitational force field.

In general, a vector field is a function whose domain is a set of points in (or ) and whose range is a set of vectors in V₂ (or V₃).

Vector Fields

The best way to picture a vector field is to draw the arrow representing the vector F(x, y) starting at the point (x, y).

Of course, it’s impossible to do this for all points (x, y), but we can gain a reasonable impression of F by doing it for a few representative points in D as in Figure 3.

Vector field on

Vector Fields

Since F(x, y) is a two-dimensional vector, we can write it in terms of its component functions P and Q as follows:

F(x, y) = P(x, y) i + Q(x, y) j =

〈

〉

or, for short, F = P i + Q j

Notice that P and Q are scalar functions of two variables and are sometimes called scalar fields to distinguish them from vector fields.

Vector Fields

A vector field F on is pictured in Figure 4.

We can express it in terms of its component functions P, Q, and R as

F(x, y, z) = P(x, y, z) i + Q(x, y, z) j + R(x, y, z) k

Figure 4

Vector field on

Vector Fields

As with the vector functions, we can define continuity of vector fields and show that F is continuous if and only if its component functions P, Q, and R are continuous.

We sometimes identify a point (x, y, z) with its position vector x =

〈

〉

Then F becomes a function that assigns a vector F(x) to a vector x.

Example 1

A vector field on is defined by F(x, y) = –y i + x j.

Describe F by sketching some of the vectors F(x, y) as in Figure 3.

Figure 3

Vector field on

Example 1 – Solution

Since F(1, 0) = j, we draw the vector j =

〈

〉

F(x, y) = –y i + x j

Example 1 – Solution

Continuing in this way, we calculate several other

representative values of F(x, y) in the table and draw the corresponding vectors to represent the vector field in Figure 5.

cont’d

Example 1 – Solution

It appears from Figure 5 that each arrow is tangent to a circle with center the origin.

cont’d

Example 1 – Solution

To confirm this, we take the dot product of the position vector x = x i + y j with the vector F(x) = F(x, y):

x  F(x) = (x i + y j)  (–y i + x j)

= –xy + yx

= 0

This shows that F(x, y) is perpendicular to the position

vector

〈

〉

cont’d

Example 1 – Solution

Notice also that

so the magnitude of the vector F(x, y) is equal to the radius

cont’d

Example 3

Imagine a fluid flowing steadily along a pipe and let V(x, y, z) be the velocity vector at a point (x, y, z).

Then V assigns a vector to each point (x, y, z) in a certain domain E (the interior of the pipe) and so V is a vector field on called a velocity field.

A possible velocity field is illustrated in Figure 13.

Velocity field in fluid flow

Example 3

The speed at any given point is indicated by the length of the arrow.

Velocity fields also occur in other areas of physics.

For instance, the vector field in Example 1 could be used as the velocity field describing the counterclockwise

rotation of a wheel.

cont’d

Example 4

Newton’s Law of Gravitation states that the magnitude of

the gravitational force between two objects with masses m and M is

where r is the distance between the objects and G is the gravitational constant. (This is an example of an inverse square law.)

Let’s assume that the object with mass M is located at the origin in . (For instance, M could be the mass of the earth and the origin would be at its center.)

Example 4

Let the position vector of the object with mass m be x =

〈

〉

The gravitational force exerted on this second object acts toward the origin, and the unit vector in this direction is

Therefore the gravitational force acting on the object at x =

〈

〉

cont’d

Example 4

[Physicists often use the notation r instead of x for the position vector, so you may see Formula 3 written in the form F = –(mMG/r³)r.]

The function given by Equation 3 is an example of a vector field, called the gravitational field, because it associates a vector [the force F(x)] with every point x in space.

cont’d

Example 4

Formula 3 is a compact way of writing the gravitational field, but we can also write it in terms of its component functions by using the facts that x = x i + y j + z k and

| x | = :

cont’d

Example 4

The gravitational field F is pictured in Figure 14.

Figure 14

Gravitational force field

cont’d

Example 5

Suppose an electric charge Q is located at the origin.

According to Coulomb’s Law, the electric force F(x) exerted by this charge on a charge q located at a point (x, y, z) with position vector x =

〈

〉

where ε is a constant (that depends on the units used).

For like charges, we have qQ > 0 and the force is repulsive;

Example 5

Notice the similarity between Formulas 3 and 4. Both vector fields are examples of force fields.

Instead of considering the electric force F, physicists often consider the force per unit charge:

Then E is a vector field on called the electric field of Q.

cont’d

Gradient Fields

Gradient Fields

If f is a scalar function of two variables, recall that its gradient ∇f (or grad f ) is defined by

∇f(x, y) = f_x(x, y) i + f_y(x, y) j

Therefore ∇f is really a vector field on and is called a gradient vector field.

Likewise, if f is a scalar function of three variables, its gradient is a vector field on given by

∇f(x, y, z) = f_x(x, y, z) i + f_y(x, y, z) j + f_z(x, y, z) k

Example 6

Find the gradient vector field of f(x, y) = x²y – y³. Plot the gradient vector field together with a contour map of f. How are they related?

Solution:

The gradient vector field is given by

Example 6 – Solution

Figure 15 shows a contour map of f with the gradient vector field.

Notice that the gradient vectors are perpendicular to the level curves.

Figure 15

cont’d

Example 6 – Solution

Notice also that the gradient vectors are long where the level curves are close to each other and short where the curves are farther apart.

That’s because the length of the gradient vector is the value of the directional derivative of f and closely spaced level curves indicate a steep graph.

cont’d

Gradient Fields

A vector field F is called a conservative vector field if it is the gradient of some scalar function, that is, if there exists a function f such that F = ∇f.

In this situation f is called a potential function for F.

Not all vector fields are conservative, but such fields do arise frequently in physics.

Gradient Fields

For example, the gravitational field F in Example 4 is conservative because if we define

then