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The mass of asteroid 1 is 750,000 kg and the mass of asteroid 2 is 130,000 kg. Assume asteroid 1 is located at the origin, and asteroid 2 is located at ( 15 , −5 , 10 ) , measured in units of 10 to the eighth power kilometers. Given that the universal gravitational constant is G = 6.67384 × 10 −11 m 3 kg −1 s −2 , find the gravitational force vector that asteroid 1 exerts on asteroid 2.

1.49063 × 10 −18 , 4.96876 × 10 −19 , 9.93752 × 10 −19 N

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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 = f = f x ( x , y ) i + f y ( x , y ) j .

We can see from the form in which the gradient is written that f is a vector field in 2 . Similarly, if f is a function of x , y , and z , then the gradient of f is

grad f = 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 3 .

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 2 or in 3 is a gradient field    if there exists a scalar function f such that f = F .

Sketching a gradient vector field

Use technology to plot the gradient vector field of f ( x , y ) = x 2 y 2 f ( x , y ) = x 2 y 2 .

The gradient of f is f = 2 x y 2 , 2 x 2 y f = 2 x y 2 , 2 x 2 y . To sketch the vector field, use a computer algebra system such as Mathematica. [link] shows f .

A visual representation of the given gradient vector field in two dimensions. The arrows point up above the x axis and down below the x axis, and they point left on the left side of the y axis and to the right on the right side of the y axis. The further the arrows are from zero, the more vertical they are, and the closer the arrows are to zero, the more horizontal they are.
The gradient vector field is f , where f ( x , y ) = x 2 y 2 .
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Use technology to plot the gradient vector field of f ( x , y ) = sin x cos y .


A visual representation of the given given vector in two dimensions. The arrows seem to be forming several ovals. The first is around the origin, where the arrows curve to the right above and below the x axis. The closer the arrows are to the x axis, the flatter they are. There appear to be six other ovals, three on either side of the central one. The vectors get longer as they get farther from the origin, and then they start to get shorter again.

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Consider the function f ( x , y ) = x 2 y 2 from [link] . [link] 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.

A visual representation of the given gradient field. The arrows are flatter the closer they are to the x axis and more vertical the further they are from the x axis. The arrows point left to the left of the y axis, and they point to the right to the right of the y axis. They point up above the x axis and down below the x axis. Severl level curves are drawn, each asymptotically approaching the axes. As the level curves get closer together, the magnitude of the gradient vactors increases.
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 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 f = F . This is the case for certain contexts in physics, for example.

Practice Key Terms 7

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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