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5. Vector Fields

Vector fields are an important tool for describing many physical concepts, such as gravitation and electromagnetism, which affect the behavior of objects over a large region of a plane or of space. They are also useful for dealing with large-scale behavior such as atmospheric storms or deep-sea ocean currents. In this section, we examine the basic definitions and graphs of vector fields so we can study them in more detail in the rest of this chapter.

Examples of Vector Fields

How can we model the gravitational force exerted by multiple astronomical objects? How can we model the velocity of water particles on the surface of a river? Figure 3.5.1 gives visual representations of such phenomena.

 

Figure 3.5.1(a) shows a gravitational field exerted by two astronomical objects, such as a star and a planet or a planet and a moon. At any point in the figure, the vector associated with a point gives the net gravitational force exerted by the two objects on an object of unit mass. The vectors of largest magnitude in the figure are the vectors closest to the larger object. The larger object has greater mass, so it exerts a gravitational force of greater magnitude than the smaller object.

 

Figure 3.5.1(b) shows the velocity of a river at points on its surface. The vector associated with a given point on the river’s surface gives the velocity of the water at that point. Since the vectors to the left of the figure are small in magnitude, the water is flowing slowly on that part of the surface. As the water moves from left to right, it encounters some rapids around a rock. The speed of the water increases, and a whirlpool occurs in part of the rapids.
Figure 3.5.1  (a) The gravitational field exerted by two astronomical bodies on a small object. (b) The vector velocity field of water on the surface of a river shows the varied speeds of water. Red indicates that the magnitude of the vector is greater, so the water flows more quickly; blue indicates a lesser magnitude and a slower speed of water flow.
Each figure illustrates an example of a vector field. Intuitively, a vector field is a map of vectors. In this section, we study vector fields in ℝ2 and ℝ3.

Definition

A vector field F in ℝ2 is an assignment of a two-dimensional vector F(x,y) to each point (x,y) of a subset D of ℝ2. The subset D is the domain of the vector field.
A vector field F in ℝ3 is an assignment of a two-dimensional vector F(x,y) to each point (x,y) of a subset D of ℝ3. The subset D is the domain of the vector field.

Vector Fields in ℝ2

A vector field in ℝ2 can be represented in either of two equivalent ways. The first way is to use a vector with components that are two-variable functions:
F(x, y) = \langle P(x,y), Q(x,y) \rangle
The second way is to use the standard unit vectors:
F(x, y) = P(x,y) i + Q(x,y) j
A vector field is said to be continuous if its component functions are continuous.

Drawing a Vector Field

We can now represent a vector field in terms of its components of functions or unit vectors, but representing it visually by sketching it is more complex because the domain of a vector field is in ℝ2, as is the range. Therefore the “graph” of a vector field in ℝ2 lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in ℝ2 in a plane itself. To do this, draw the vector associated with a given point at the point in a plane. For example, suppose the vector associated with point (4,-1) is \langle 3, 1 \rangle. Then, we would draw vector \langle 3, 1 \rangle at point (4,-1).

 

We should plot enough vectors to see the general shape, but not so many that the sketch becomes a jumbled mess. If we were to plot the image vector at each point in the region, it would fill the region completely and is useless. Instead, we can choose points at the intersections of grid lines and plot a sample of several vectors from each quadrant of a rectangular coordinate system in ℝ2.

 

There are two types of vector fields in ℝ2 on which this chapter focuses: radial fields and rotational fields. Radial fields model certain gravitational fields and energy source fields, and rotational fields model the movement of a fluid in a vortex. In a radial field, all vectors either point directly toward or directly away from the origin. Furthermore, the magnitude of any vector depends only on its distance from the origin. In a radial field, the vector located at point (x,y) is perpendicular to the circle centered at the origin that contains point (x,y), and all other vectors on this circle have the same magnitude.

Example: Velocity Field of a Fluid

Suppose that v(x,y) = -\frac{2y}{x^2 + y^2} i + \frac{2x}{x^2 + y^2}j is the velocity field of a fluid. How fast is the fluid moving at point (1,-1)? (Assume the units of speed are meters per second.)
Solution
To find the velocity of the fluid at point (1,-1), substitute the point into v:
v(1, -1) = -\frac{2(-1)}{1+1} i + \frac{2(1)}{1+1}j = i+j
The speed of the fluid at (1,-1) is the magnitude of this vector.
Therefore, the speed is || i+j || = \sqrt{2} m/sec.

Vector Fields in ℝ3

We have seen several examples of vector fields in ℝ2; let’s now turn our attention to vector fields in ℝ3. These vector fields can be used to model gravitational or electromagnetic fields, and they can also be used to model fluid flow or heat flow in three dimensions. A two-dimensional vector field can really only model the movement of water on a two-dimensional slice of a river (such as the river’s surface). Since a river flows through three spatial dimensions, to model the flow of the entire depth of the river, we need a vector field in three dimensions.

 

The extra dimension of a three-dimensional field can make vector fields in ℝ3 more difficult to visualize, but the idea is the same. To visualize a vector field in ℝ3, plot enough vectors to show the overall shape. We can use a similar method to visualizing a vector field in \mathbb{R}^2 by choosing points in each octant.

Just as with vector fields in ℝ2, we can represent vector fields in ℝ3 with component functions. We simply need an extra component function for the extra dimension. We write either

F(x,y,z) = \langle P(xy,z), Q(xy,z), R(xy,z) \rangle
or
F(x,y,z) = P(xy,z)i+ Q(xy,z)j+ R(xy,z)k

Example: Sketching a Vector Field in Three Dimensions

Describe vector field F(x,y,z)= \langle 1,1,z \rangle
Solution
For this vector field, the x and y components are constant, so every point in ℝ3 has an associated vector with x and y components equal to one. To visualize F, we first consider what the field looks like in the xy-plane. In the xy-plane, z=0. Hence, each point of the form (a,b,0) has vector \langle 1,1,0 \rangle associated with it. For points not in the xy-plane but slightly above it, the associated vector has a small but positive z component, and therefore the associated vector points slightly upward. For points that are far above the xy-plane, the z component is large, so the vector is almost vertical. The figure below shows this vector field.
A visual representation of vector field F(x,y,z)=(1,1,z).

Gradient Fields

In this section, we study a special kind of vector field called a gradient field or a conservative field. These vector fields are extremely important in physics because they can be used to model physical systems in which energy is conserved. Gravitational fields and electric fields associated with a static charge are examples of gradient fields.
Recall that if f is a (scalar) function of x and y, then the gradient of f is
grad f = \nabla f = f_x (x,y) i + f_y (x,y,z) j

 

We can see from the form in which the gradient is written that \nabla f is a vector field in ℝ2. Similarly, if f is a function of x, y, and z, then the gradient of f is
grad f = \nabla f = f_x (x,y,z) i + f_y (x,y,z) j + f_z (x,y,z) k

The gradient of a three-variable function is a vector field in ℝ3.

A gradient field is a vector field that can be written as the gradient of a function, and we have the following definition.

Definition

A vector field F in ℝ2 or in ℝ3 is a gradient field if there exists a scalar function f such that \nabla f = F.

Example: Sketching a Gradient Vector Field

Use technology to plot the gradient vector field of f(x,y)=x_2 y_2.
Solution
The gradient of f is \nabla f= \langle2xy^2,2x^2y \rangle. To sketch the vector field, use a computer algebra system such as Mathematica. The figure below shows \nabla f.
The gradient vector field is grad f, where f(x,y)=x^2y^2.
Consider the function f(x,y)=x^2 y^2 from example above. Figure 3.5.2 shows the level curves of this function overlaid on the function’s gradient vector field. The gradient vectors are perpendicular to the level curves, and the magnitudes of the vectors get larger as the level curves get closer together, because closely grouped level curves indicate the graph is steep, and the magnitude of the gradient vector is the largest value of the directional derivative. Therefore, you can see the local steepness of a graph by investigating the corresponding function’s gradient field.
Figure 3.5.2 The gradient field of f(x,y)=x^2 y^2 and several level curves of f. Notice that as the level curves get closer together, the magnitude of the gradient vectors increases.
As we learned earlier, a vector field F is a conservative vector field, or a gradient field if there exists a scalar function f such that \nabla f = F. In this situation, f is called a potential function for F. Conservative vector fields arise in many applications, particularly in physics. The reason such fields are called conservative is that they model forces of physical systems in which energy is conserved. We study conservative vector fields in more detail later in this chapter.

 

You might notice that, in some applications, a potential function f for F is defined instead as a function such that -\nabla f=F. This is the case for certain contexts in physics, for example.

Example: Verifying a Potential Function

Is f (x,y,z)= x^2 yz-\sin(xy) a potential function for vector field
F(x,y,z)=\langle 2xyz-y cos(xy), x^2 z-x \cos(xy), x^2 y \rangle?
Solution
We need to confirm whether \nabla f=F.  We have
f_x=2xyz-y \cos(xy), f_y=x^2 z-x \cos(xy), and f_z=x^2 y.
Therefore, \nabla f=F and f is a potential function for F.

If F is a conservative vector field, then there is at least one potential function f such that \nabla f=F. But, could there be more than one potential function? If so, is there any relationship between two potential functions for the same vector field? Before answering these questions, let’s recall some facts from single-variable calculus to guide our intuition. Recall that if k(x) is an integrable function, then k has infinitely many antiderivatives. Furthermore, if F and G are both antiderivatives of k, then F and G differ only by a constant. That is, there is some number C such that F(x)=G(x)+C.

 

Now let F be a conservative vector field and let f and g be potential functions for F. Since the gradient is like a derivative, F being conservative means that F is “integrable” with “antiderivatives” f and g. Therefore, if the analogy with single-variable calculus is valid, we expect there is some constant C such that f(x)=g(x)+C. The next theorem says that this is indeed the case.
To state the next theorem with precision, we need to assume the domain of the vector field is connected and open. To be connected means if P1 and P2 are any two points in the domain, then you can walk from P1 to P2 along a path that stays entirely inside the domain.

Theorem

Uniqueness of Potential Functions
Let F be a conservative vector field on an open and connected domain and let f and g be functions such that \nabla f=F and \nabla g=F. Then, there is a constant C such that f=g+C.
Conservative vector fields also have a special property called the cross-partial property. This property helps test whether a given vector field is conservative.

Theorem

The Cross-Partial Property of Conservative Vector Fields
Let F be a vector field in two or three dimensions such that the component functions of F have continuous second-order mixed-partial derivatives on the domain of F.
If F(x,y)=\langle P(x,y),Q(x,y) \rangle is a conservative vector field in ℝ2, then
\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}.
If F(x,y)=\langle P(x,y,z), Q(x,y,z), R(x,y,z) \rangle is a conservative vector field in ℝ3, then
\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}, \frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y}, and \frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}

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