**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 V*_{2} *(or V*_{3}).

### 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 = **

### 〈

*P(x, y), Q(x, y)*

### 〉

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 = **

### 〈

^{x, y, z}### 〉

**and write F(x) instead of F(x, y, z).****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 = **

### 〈

0, 1### 〉

starting at the point (1, 0) in Figure 5.**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

### 〈

^{x, y}### 〉

and is therefore tangent to a circle with center the origin and radiuscont’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 = **

### 〈

^{x, y, z}### 〉

**. Then r = | x |, so r**^{2}

**= | x |**

^{2}.

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 = **

### 〈

^{x, y, z}### 〉

^{is}

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/r**^{3}**)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 = **

### 〈

^{x, y, z}### 〉

^{is}

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) = x*^{2}*y – y*^{3}. 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