6.1 Vector fields  (Page 5/15)

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

We can see from the form in which the gradient is written that $\text{∇}f$ is a vector field in ${ℝ}^2.$ Similarly, if $f$ is a function of x , y , and z , then the gradient of $f$ is

The gradient of a three-variable function is a vector field in ${ℝ}^3.$

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

Sketching a gradient vector field

Use technology to plot the gradient vector field of $f\left(x,y\right)=x^2{y}^{2}f\left(x,y\right)=x^2{y}^2.$

The gradient of $f$ is $\text{∇}f=⟨2x{y}^2,2x^{2}y⟩\text{∇}f=⟨2x{y}^2,2x^{2}y⟩.$ To sketch the vector field, use a computer algebra system such as Mathematica. [link] shows $\text{∇}f.$

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Consider the function $f\left(x,y\right)=x^2{y}^2$ 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.

As we learned earlier, a vector field $\text{F}$ is a conservative vector field, or a gradient field if there exists a scalar function $f$ such that $\text{∇}f=\text{F}.$ In this situation, $f$ is called a potential function    for $\text{F}.$ 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 $\text{−}\text{∇}f=\text{F}.$ This is the case for certain contexts in physics, for example.