Vector fields, line integrals and the big theorems

Vector Fields

This section uses tools from the chapter on multivariable functions and their derivatives, with specific references to gradients. Additionally, vector calculations and geometry are used throughout to understand the output of the vector field.

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? gives visual representations of such phenomena.

(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.

(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.

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 \({ℝ}^{2}\) and \({ℝ}^{3}.\)

Condensed — the full section is in OpenStax Calculus Volume 3.

Introduction

Vectors have played a central role in our study of multivariable calculus. We know how to do operations on vectors (addition, scalar multiplication, dot product, etc.), and we have seen how vectors can be used to describe curves in \(\R^2\) and \(\R^3\). The examples of using vectors to describe curves was our first example of a vector-valued function. In a curve is traced by the terminal point of \(\vr(t)\), a function that has a real number as an input and produces a vector in \(\R^2\) or \(\R^3\). In this section, we will expand our understanding of vector-valued functions to take a point \((x,y)\) in \(\R^2\) (or a point \((x,y,z)\) in \(\R^3\)) as an input and produce a vector (typically in \(\R^2\) or \(\R^3\), respectively) as output.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

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 \({ℝ}^{2},\) as is the range. Therefore the “graph” of a vector field in \({ℝ}^{2}\) lives in four-dimensional space. Since we cannot represent four-dimensional space visually, we instead draw vector fields in \({ℝ}^{2}\) 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 \(〈3,1〉.\) Then, we would draw vector \(〈3,1〉\) 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 \({ℝ}^{2}.\)

There are two types of vector fields in \({ℝ}^{2}\) 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

Try it.

Sketch the vector field \(\text{F}(x,y)=\frac{x}{2}\ \text{i}+\frac{y}{2}\ \text{j}.\)

Solution

To sketch this vector field, choose a sample of points from each quadrant and compute the corresponding vector. The following table gives a representative sample of points in a plane and the corresponding vectors.

\((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)
\((1,0)\)	\(〈\frac{1}{2},0〉\)	\((2,0)\)	\(〈1,0〉\)	\((1,1)\)	\(〈\frac{1}{2},\frac{1}{2}〉\)
\((0,1)\)	\(〈0,\frac{1}{2}〉\)	\((0,2)\)	\(〈0,1〉\)	\((-1,1)\)	\(〈-\frac{1}{2},\frac{1}{2}〉\)
\((-1,0)\)	\(〈-\frac{1}{2},0〉\)	\((-2,0)\)	\(〈-1,0〉\)	\((-1,-1)\)	\(〈-\frac{1}{2},-\frac{1}{2}〉\)
\((0,-1)\)	\(〈0,-\frac{1}{2}〉\)	\((0,-2)\)	\(〈0,-1〉\)	\((1,-1)\)	\(〈\frac{1}{2},-\frac{1}{2}〉\)

(a) shows the vector field. To see that each vector is perpendicular to the corresponding circle, (b) shows circles overlain on the vector field.

Condensed — the full section is in OpenStax Calculus Volume 3.

Examples of Vector Fields

As showed, a velocity vector field is an example of a scenario where associating a vector to each point in a region is useful. We denote such a vector field by \(\vF(x,y)\) or \(\vF(x,y,z)\), where the vector associated to the point \((x,y)\) or \((x,y,z)\) is the velocity of something at that point. Wind velocity is one example, but another example would be the velocity of a flowing fluid. shows such a velocity vector field. Technically, it only shows some of the vectors in the vector field, since the figure would be unintelligible if all of the vectors were shown. This is illustrated by the inset in the upper left corner, which gives a better picture of what we would see if we zoomed in on the red square of the main figure.

Force fields, such as those created by gravity, are also examples of vector fields. For example, the earth exerts a gravitational force on objects which is directed from the center of the object to the center of the earth. The magnitude of the force vector is determined by the distance between the object and the earth (by an reciprocal squared relationship.) An illustration of this vector field can be seen in , where the earth is positioned at the origin, but not shown. Notice that the vectors get shorter as the distance from the origin increases, reflecting the fact that the gravitational force is weaker at larger distances from the origin (Earth).

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

\[\text{grad}\ f=\text{∇}f={f}_{x}(x,y)\text{i}+{f}_{y}(x,y)\text{j}.\]

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

\[\text{grad}\ f=\text{∇}f={f}_{x}(x,y,z)\text{i}+{f}_{y}(x,y,z)\text{j}+{f}_{z}(x,y,z)\text{k}.\]

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.

Example

Try it.

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

Solution

The gradient of \(f\) is \(\text{∇}f=〈2x{y}^{2},2{x}^{2}y〉.\) To sketch the vector field, use a computer algebra system such as Mathematica. shows \(\text{∇}f.\)

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

Condensed — the full section is in OpenStax Calculus Volume 3.

Mathematical Vector Fields

As suggested in the introduction and , vector fields can be specified using the notation of functions and vectors.

Since \(\vF(x,y,z)\) is a vector, it has \(\vi\), \(\vj\), and \(\vk\) components. Each of these components is a scalar function of the point \((x,y,z)\), and so we will often write \[\begin{aligned}\end{aligned}\]

For example, if \(\vF(x,y,z) = \langle x^2,xy\sin(z),y^3\rangle\), then the component functions of \(\vF\) would be \(F_1(x,y,z) = x^2\), \(F_2(x,y,z) = xy\sin(z)\), and \(F_3(x,y,z) = y^3\). Any time we are considering a vector field \(\vF(x,y,z)\), the definitions of functions \(F_1\), \(F_2\), and \(F_3\) should be inferred in this manner. (For a vector field \(\vF(x,y)\) in \(2\)-space, we only have the functions \(F_1\) and \(F_2\), which are defined analogously.)

Key Concepts

• A vector field assigns a vector \(\text{F}(x,y)\) to each point \((x,y)\) in a subset D of \({ℝ}^{2}\ \text{or}\ {ℝ}^{3}.\) \(\text{F}(x,y,z)\) to each point \((x,y,z)\) in a subset D of \({ℝ}^{3}.\)
• Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.
• We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.
• A vector field \(\text{F}\) is called conservative if there exists a scalar function \(f\) such that \(\text{∇}f=\text{F}.\)

Plotting Vector Fields

gave you a chance to plot some vectors in the vector fields \(\vF(x,y) = \langle y,x\rangle\) and \(\vG(x,y) = \langle 0,-x\rangle\). It would be impossible to sketch all of the vectors in these vector fields, since there is one for every point in the plane. In fact, even sketching many more of the vectors than you were asked to in the preview activity rapidly becomes tedious. Fortunately, computers can do a great job of making such sketches. One thing to keep in mind, however, is that the magnitudes of the vectors in computer plots are typically scaled, including plots of vector fields we will encounter later in this text. To illustrate this, consider the two plots of the vector field \(\vF(x,y) = y\vi + x\vj\) in .

The left plot shows some of the vectors and accurately depicts all of their magnitudes, making the figure very hard to understand, especially along the lines \(y=x\) and \(y=-x\). The plot on the right, however, uses a uniform rescaling to make the figure easier to read. As before, each vector's direction is completely accurate, but now the magnitudes are much smaller. However, the relative magnitudes are preserved, helping us to see that vectors farther from the origin have larger magnitude than those closer to the origin.

Activity

The plot in illustrates the vector field \(\vF(x,y) = y\vi -x\vj\).

Starting with one of the vectors near the point \((2,0)\), sketch a curve that follows the direction of the vector field \(\vF\). To help visualize what you are doing, it may be useful to think of the vector field as the velocity vector field for some flowing water and that you are imagining tracing the path that a tiny particle inserted into the water would follow as the water moves it around.

Repeat the previous step for at least two other starting points not on the curve you previously sketched.

What shape do the curves you sketched in the previous two steps form?

Verify that \(\vF(x,y)\) is orthogonal to \(\langle x,y\rangle\).

Calculate the gradient of the function \(f(x,y) = x^2 + y^2\) and write a sentence comparing your result to the vector \(x\vi + y\vj\).

Write a sentence describing the geometric relationship between \(\vF(x,y)\) and a circle centered at the origin. What is the relationship between \(\vecmag{\vF(x,y)}\) and the radius of that circle?

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Key Equations

Vector field in \({ℝ}^{2}\)	\(\text{F}(x,y)=〈P(x,y),Q(x,y)〉\) or \(\text{F}(x,y)=P(x,y)\text{i}+Q(x,y)\text{j}\)
Vector field in \({ℝ}^{3}\)	\(\text{F}(x,y,z)=〈P(x,y,z),Q(x,y,z),R(x,y,z)〉\) or \(\text{F}(x,y,z)=P(x,y,z)\text{i}+Q(x,y,z)\text{j}+R(x,y,z)\text{k}\)

Gradient Vector Fields

Without using the terminology, we've actually already encountered one very important family of vector fields a number of times. Given a function \(f\) of two or three (or more!) variables, the gradient of \(f\) is a vector field, since for any point where \(f\) has first-order partial derivatives, \(\grad{f}\) assigns a vector to that point (look at for a review).

Activity

In there are three sets of axes showing level curves for functions \(f\), \(g\), and \(h\), respectively. Sketch at least six vectors in the gradient vector field for each function. In making your sketches, you don't have to worry about getting vector magnitudes precise, but you should ensure that the relative magnitudes (and directions) are correct for each function independently.

Verify that \(\vF(x,y) = \langle 6xy,3x^2+9\sqrt{y}\rangle\) is a gradient vector field by finding a function \(f\) such that \(\nabla f(x,y) = \vF(x,y)\). For reasons originating in physics, such a function \(f\) is called a potential function for the vector field \(\vF\).

Is the function \(f\) found in part unique? That is, can you find another function \(g\) such that \(\nabla g(x,y)= \vF(x,y)\) but \(f\neq g\)?

Is the vector field \(\vF(x,y) = 6xy\vi +(2x+9\sqrt{y})\vj\) a gradient vector field? Why or why not?

In this text, we will most often refer to a vector field that can be written as the gradient of some scalar valued function to be a gradient field. In other sources and fields, a vector field \(\vF\) where \(\vF=\nabla f\) for some scalar valued \(f\) is called a conservative vector field. We will describe this term more in but mention this synonym term here as well.

Vector Fields

For the following exercises, determine whether the statement is true or false.

For the following exercises, describe each vector field by drawing some of its vectors.

For the following exercises, find the gradient vector field of each function \(f.\)

For the following exercises, write formulas for the vector fields with the given properties.

For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function \(V(x,y)=c\ \text{ln}(\frac{{r}_{0}}{\sqrt{{x}^{2}+{y}^{2}}}),\) where \(c>0\) is a constant and \({r}_{0}\) is a reference distance at which the potential is assumed to be zero.

A flow line (or streamline) of a vector field \(\text{F}\) is a curve \(\text{r}(t)\) such that \(d\text{r}\text{/}dt=\text{F}(\text{r}(t)).\) If \(\text{F}\) represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve \(\text{c}(t)\) is a flow line of the given velocity vector field \(\text{F}(x,y,z).\)

For the following exercises, let \(\text{F}=x\text{i}+y\text{j},\) \(\text{G}=\text{-}y\text{i}+x\text{j},\) and \(\text{H}=x\text{i}-y\text{j}.\) Match F, G, and H with their graphs.

1. \(\text{F}+\text{G}\)
2. \(\text{F}+\text{H}\)
3. \(\text{G}+\text{H}\)
4. \(\text{-}\text{F}+\text{G}\)

Condensed — the full section is in OpenStax Calculus Volume 3.

Toimiva esimerkki: gradient of x^2 + y^2

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Let \(\text{F}(x,y)=(2{y}^{2}+x-4)\text{i}+\text{cos}(x)\text{j}\) be a vector field in \({ℝ}^{2}.\) Note that this is an example of a continuous vector field since both component functions are continuous. What vector is associated with point \((0,-1)?\)

   Paljasta vastaus

   Substitute the point values for x and y:

   \[\begin{array}{ll}\text{F}(0,-1) & =(2{(-1)}^{2}+0-4)\text{i}+\text{cos}(0)\text{j} \\ & =-2\text{i}+\text{j}.\end{array}\]

2. Let \(\text{G}(x,y)={x}^{2}y\text{i}-(x+y)\text{j}\) be a vector field in \({ℝ}^{2}.\) What vector is associated with the point \((-2,3)?\)

   Paljasta vastaus

   \(12\text{i}-\text{j}\)

3. Sketch the vector field \(\text{F}(x,y)=\frac{x}{2}\ \text{i}+\frac{y}{2}\ \text{j}.\)

   Paljasta vastaus

   To sketch this vector field, choose a sample of points from each quadrant and compute the corresponding vector. The following table gives a representative sample of points in a plane and the corresponding vectors.

   \((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)
   \((1,0)\)	\(〈\frac{1}{2},0〉\)	\((2,0)\)	\(〈1,0〉\)	\((1,1)\)	\(〈\frac{1}{2},\frac{1}{2}〉\)
   \((0,1)\)	\(〈0,\frac{1}{2}〉\)	\((0,2)\)	\(〈0,1〉\)	\((-1,1)\)	\(〈-\frac{1}{2},\frac{1}{2}〉\)
   \((-1,0)\)	\(〈-\frac{1}{2},0〉\)	\((-2,0)\)	\(〈-1,0〉\)	\((-1,-1)\)	\(〈-\frac{1}{2},-\frac{1}{2}〉\)
   \((0,-1)\)	\(〈0,-\frac{1}{2}〉\)	\((0,-2)\)	\(〈0,-1〉\)	\((1,-1)\)	\(〈\frac{1}{2},-\frac{1}{2}〉\)

   (a) shows the vector field. To see that each vector is perpendicular to the corresponding circle, (b) shows circles overlain on the vector field.

4. Draw the radial field \(\text{F}(x,y)=-\frac{x}{3}\ \text{i}-\frac{y}{3}\ \text{j}.\)

   Paljasta vastaus

   
   

5. Sketch the vector field \(\text{F}(x,y)=〈y,\text{-}x〉.\)

   Paljasta vastaus

   Create a table (see the one that follows) using a representative sample of points in a plane and their corresponding vectors. shows the resulting vector field.

   \((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)
   \((1,0)\)	\(〈0,-1〉\)	\((2,0)\)	\(〈0,-2〉\)	\((1,1)\)	\(〈1,-1〉\)
   \((0,1)\)	\(〈1,0〉\)	\((0,2)\)	\(〈2,0〉\)	\((-1,1)\)	\(〈1,1〉\)
   \((-1,0)\)	\(〈0,1〉\)	\((-2,0)\)	\(〈0,2〉\)	\((-1,-1)\)	\(〈-1,1〉\)
   \((0,-1)\)	\(〈-1,0〉\)	\((0,-2)\)	\(〈-2,0〉\)	\((1,-1)\)	\(〈-1,-1〉\)

6. Sketch vector field \(\text{F}(x,y)=\frac{y}{{x}^{2}+{y}^{2}}\ \text{i}-\frac{x}{{x}^{2}+{y}^{2}}\ \text{j}.\)

   Paljasta vastaus

   To visualize this vector field, first note that the dot product \(\text{F}(a,b)\cdot (a\text{i}+b\text{j})\) is zero for any point \((a,b).\) Therefore, each vector is tangent to the circle on which it is located. Also, as \((a,b)\to (0,0),\) the magnitude of \(\text{F}(a,b)\) goes to infinity. To see this, note that

   \[||\text{F}(a,b)||=\sqrt{\frac{{a}^{2}+{b}^{2}}{{({a}^{2}+{b}^{2})}^{2}}}=\sqrt{\frac{1}{{a}^{2}+{b}^{2}}}.\]

   Since \(\frac{1}{{a}^{2}+{b}^{2}}\to \infty\) as \((a,b)\to (0,0),\) then \(||\text{F}(a,b)||\to \infty\) as \((a,b)\to (0,0).\) This vector field looks similar to the vector field in , but in this case the magnitudes of the vectors close to the origin are large. The table below shows a sample of points and the corresponding vectors, and shows the vector field. Note that this vector field models the whirlpool motion of the river in (b). The domain of this vector field is all of \({ℝ}^{2}\) except for point \((0,0).\)

   \((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)	\((x,y)\)	\(\text{F}(x,y)\)
   \((1,0)\)	\(〈0,-1〉\)	\((2,0)\)	\(〈0,-\frac{1}{2}〉\)	\((1,1)\)	\(〈\frac{1}{2},-\frac{1}{2}〉\)
   \((0,1)\)	\(〈1,0〉\)	\((0,2)\)	\(〈\frac{1}{2},0〉\)	\((-1,1)\)	\(〈\frac{1}{2},\frac{1}{2}〉\)
   \((-1,0)\)	\(〈0,1〉\)	\((-2,0)\)	\(〈0,\frac{1}{2}〉\)	\((-1,-1)\)	\(〈-\frac{1}{2},\frac{1}{2}〉\)
   \((0,-1)\)	\(〈-1,0〉\)	\((0,-2)\)	\(〈-\frac{1}{2},0〉\)	\((1,-1)\)	\(〈-\frac{1}{2},-\frac{1}{2}〉\)

7. Sketch vector field \(\text{F}(x,y)=〈-2y,2x〉.\) Is the vector field radial, rotational, or neither?

   Paljasta vastaus

   Rotational
   

8. Suppose that \(\text{v}(x,y)=-\frac{2y}{{x}^{2}+{y}^{2}}\ \text{i}+\frac{2x}{{x}^{2}+{y}^{2}}\ \text{j}\) 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.)

   Paljasta vastaus

   To find the velocity of the fluid at point \((1,-1),\) substitute the point into v:

   \[\text{v}(1,-1)=-\frac{2(-1)}{1+1}\ \text{i}+\frac{2(1)}{1+1}\ \text{j}=\text{i}+\text{j}.\]

   The speed of the fluid at \((1,-1)\) is the magnitude of this vector. Therefore, the speed is \(||\text{i}+\text{j}||=\sqrt{2}\) m/sec.

9. Vector field \(v(x,y)=〈4|x|,1〉\) models the velocity of water on the surface of a river. What is the speed of the water at point \((2,3)?\) Use meters per second as the units.

   Paljasta vastaus

   \(\sqrt{65}\) m/sec

10. Show that the vector field \(\text{G}(x,y)=〈\frac{y}{\sqrt{{x}^{2}+{y}^{2}}},-\frac{x}{\sqrt{{x}^{2}+{y}^{2}}}〉\) is a unit vector field.

    Paljasta vastaus

    To show that G is a unit field, we must show that the magnitude of each vector is 1. Note that

    \[\begin{array}{ll}\sqrt{{(\frac{y}{\sqrt{{x}^{2}+{y}^{2}}})}^{2}+{(-\frac{x}{\sqrt{{x}^{2}+{y}^{2}}})}^{2}} & =\sqrt{\frac{{y}^{2}}{{x}^{2}+{y}^{2}}+\frac{{x}^{2}}{{x}^{2}+{y}^{2}}} \\ & =\sqrt{\frac{{x}^{2}+{y}^{2}}{{x}^{2}+{y}^{2}}} \\ & =1.\end{array}\]

    Therefore, G is a unit vector field.

11. Is vector field \(\text{F}(x,y)=〈\text{-}y,x〉\) a unit vector field?

    Paljasta vastaus

    No.

12. Describe vector field \(\text{F}(x,y,z)=〈1,1,z〉.\)

    Paljasta vastaus

    For this vector field, the x and y components are constant, so every point in \({ℝ}^{3}\) 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 \(〈1,1,0〉\) 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. shows this vector field.

13. Sketch vector field \(\text{G}(x,y,z)=〈2,\frac{z}{2},1〉.\)

    Paljasta vastaus

    
    

14. Newton’s law of gravitation states that \(\text{F}=\text{-}G\ \frac{{m}_{1}{m}_{2}}{{r}^{2}}\overset{\wedge}{r},\) where G is the universal gravitational constant. It describes the gravitational field exerted by an object (object 1) of mass \({m}_{1}\) located at the origin on another object (object 2) of mass \({m}_{2}\) located at point \((x,y,z).\) Field F denotes the gravitational force that object 1 exerts on object 2, r is the distance between the two objects, and \(\overset{\wedge}{r}\) indicates the unit vector from the first object to the second. The minus sign shows that the gravitational force attracts toward the origin; that is, the force of object 1 is attractive. Sketch the vector field associated with this equation.

    Paljasta vastaus

    Since object 1 is located at the origin, the distance between the objects is given by \(r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}.\) The unit vector from object 1 to object 2 is \(\overset{\wedge}{r}=\frac{〈x,y,z〉}{||〈x,y,z〉||},\) and hence \(\overset{\wedge}{r}=〈\frac{x}{r},\frac{y}{r},\frac{z}{r}〉.\) Therefore, gravitational vector field F exerted by object 1 on object 2 is

    \[\text{F}=\text{-}G{m}_{1}{m}_{2}〈\frac{x}{{r}^{3}},\frac{y}{{r}^{3}},\frac{z}{{r}^{3}}〉.\]

    This is an example of a radial vector field in \({ℝ}^{3}.\)

    shows what this gravitational field looks like for a large mass at the origin. Note that the magnitudes of the vectors increase as the vectors get closer to the origin.

15. 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 \(G=6.67384\ \times \ {10}^{-11}{\ \text{m}}^{3}{\text{kg}}^{-1}{\text{s}}^{-2},\) find the gravitational force vector that asteroid 1 exerts on asteroid 2.

    Paljasta vastaus

    \(-1.49063\ \times \ {10}^{-18},4.96876\ \times \ {10}^{-19},-9.93752\ \times \ {10}^{-19}\text{N}\)

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

    Paljasta vastaus

    The gradient of \(f\) is \(\text{∇}f=〈2x{y}^{2},2{x}^{2}y〉.\) To sketch the vector field, use a computer algebra system such as Mathematica. shows \(\text{∇}f.\)

17. Use technology to plot the gradient vector field of \(f(x,y)=\text{sin}\ x\ \text{cos}\ y.\)

    Paljasta vastaus

    
    

18. Is \(f(x,y,z)={x}^{2}yz-\text{sin}(xy)\) a potential function for vector field

    \[\text{F}(x,y,z)=〈2xyz-y\ \text{cos}(xy),{x}^{2}z-x\ \text{cos}(xy),{x}^{2}y〉?\]
    Paljasta vastaus

    We need to confirm whether \(\text{∇}f=\text{F}.\) We have

    \[{f}_{x}=2xyz-y\ \text{cos}(xy),{f}_{y}={x}^{2}z-x\ \text{cos}(xy),\ \text{and}\ {f}_{z}={x}^{2}y.\]

    Therefore, \(\text{∇}f=\text{F}\) and \(f\) is a potential function for \(\text{F}.\)

19. Is \(f(x,y,z)={x}^{2}\text{cos}(yz)+{y}^{2}{z}^{2}\) a potential function for \(\text{F}(x,y,z)=〈2x\ \text{cos}(yz),\text{-}{x}^{2}z\ \text{sin}(yz)+2y{z}^{2},{y}^{2}〉?\)

    Paljasta vastaus

    No

20. The velocity of a fluid is modeled by field \(\text{v}(x,y)=〈xy,\frac{{x}^{2}}{2}-y〉.\) Verify that \(f(x,y)=\frac{{x}^{2}y}{2}-\frac{{y}^{2}}{2}\) is a potential function for v.

    Paljasta vastaus

    To show that \(f\) is a potential function, we must show that \(\text{∇}f=\text{v}.\) Note that \({f}_{x}=xy\) and \({f}_{y}=\frac{{x}^{2}}{2}-y.\) Therefore, \(\text{∇}f=〈xy,\frac{{x}^{2}}{2}-y〉\) and \(f\) is a potential function for v ().

21. Verify that \(f(x,y)={x}^{3}{y}^{2}+1\) is a potential function for velocity field \(\text{v}(x,y)=⟨3{x}^{2}{y}^{2},2{x}^{3}y⟩.\)

    Paljasta vastaus

    \(\text{∇}f=\text{v}\)

22. Show that rotational vector field \(\text{F}(x,y)=〈y,\text{-}x〉\) is not conservative.

    Paljasta vastaus

    Let \(P(x,y)=y\ \text{and}\ Q(x,y)=\text{-}x.\) If F is conservative, then the cross-partials would be equal—that is, \({P}_{y}\) would equal \({Q}_{x.}\) Therefore, to show that F is not conservative, check that \({P}_{y}\ne {Q}_{x}.\) Since \({P}_{y}=1\) and \({Q}_{x}=-1,\) the vector field is not conservative.

23. Show that the vector field \(\text{F}(x,y)=y\text{i}-{x}^{2}xy\text{j}\) is not conservative.

    Paljasta vastaus

    \({P}_{y}=x\ne {Q}_{x}=-2xy\)

24. Is vector field \(\text{F}(x,y,z)=〈7,-2,{x}^{3}〉\) conservative?

    Paljasta vastaus

    Let \(P(x,y,z)=7,\) \(Q(x,y,z)=-2,\) and \(R(x,y,z)={x}^{3}.\) If F is conservative, then all three cross-partial equations will be satisfied—that is, if F is conservative, then \({P}_{y}\) would equal \({Q}_{x},{Q}_{z}\) would equal \({R}_{y},\) and \({R}_{x}\) would equal \({P}_{z}.\) Note that \({P}_{y}={Q}_{x}={R}_{y}={Q}_{z}=0,\) so the first two necessary equalities hold. However, \({R}_{x}=3{x}^{2}\) and \({P}_{z}=0\) so \({R}_{x}\ne {P}_{z}.\) Therefore, \(\text{F}\) is not conservative.

25. Is vector field \(G(x,y,z)=〈y,x,xyz〉\) conservative?

    Paljasta vastaus

    No

26. The domain of vector field \(\text{F}=\text{F}(x,y)\) is a set of points \((x,y)\) in a plane, and the range of F is a set of what in the plane?

    Paljasta vastaus

    Vectors

27. Vector field \(\text{F}=〈3{x}^{2},1〉\) is a gradient field for both \({ϕ}_{1}(x,y)={x}^{3}+y\) and \({ϕ}_{2}(x,y)=y+{x}^{3}+100.\)

28. Vector field \(\text{F}=\frac{〈y,x〉}{\sqrt{{x}^{2}+{y}^{2}}}\) is constant in direction and magnitude on a unit circle.

    Paljasta vastaus

    False

29. Vector field \(\text{F}=\frac{〈y,x〉}{\sqrt{{x}^{2}+{y}^{2}}}\) is neither a radial field nor a rotation.

30. [T] \(\text{F}(x,y)=x\text{i}+y\text{j}\)

    Paljasta vastaus

    
    

31. [T] \(\text{F}(x,y)=\text{-}y\text{i}+x\text{j}\)

32. [T] \(\text{F}(x,y)=x\text{i}-y\text{j}\)

    Paljasta vastaus

    
    

33. [T] \(\text{F}(x,y)=\text{i}+\text{j}\)

34. [T] \(\text{F}(x,y)=2x\text{i}+3y\text{j}\)

    Paljasta vastaus

    
    

35. [T] \(\text{F}(x,y)=3\text{i}+x\text{j}\)

36. [T] \(\text{F}(x,y)=y\text{i}+\text{sin}\ x\text{j}\)

    Paljasta vastaus

    
    

37. [T] \(\text{F}(x,y,z)=x\text{i}+y\text{j}+z\text{k}\)

38. [T] \(\text{F}(x,y,z)=2x\text{i}-2y\text{j}-2z\text{k}\)

    Paljasta vastaus

    
    

39. [T] \(\text{F}(x,y,z)=\frac{y}{z}\ \text{i}-\frac{x}{z}\ \text{j}\)

40. \(f(x,y)=x\ \text{sin}\ y+\text{cos}\ y\)

    Paljasta vastaus

    \(\text{F}(x,y)=\text{sin}(y)\text{i}+(x\ \text{cos}\ y-\text{sin}\ y)\text{j}\)

Symbols used here

\frac{\partial f}{\partial x}

partial derivative
Derivative with respect to x, holding the other variables fixed.

\nabla f

gradient (nabla, del)
Vector of partial derivatives; points uphill.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

matrix
A rectangular array of numbers; a linear map.

\neg,\ \wedge,\ \vee,\ \Rightarrow,\ \Leftrightarrow

not, and, or, implies, iff
Logical connectives.

i

imaginary unit
i² = −1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

\iint,\ \oint

double / contour integral
Integral over a region of the plane; integral around a closed curve.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|

dot product, norm
Σ u_i v_i; the length of v, √(v·v).

How to: Vector fields, line integrals and the big theorems

1. Recognize a vector field in a plane or in space.
2. Sketch a vector field from a given equation.
3. Identify a conservative field and its associated potential function.
4. A vector field assigns a vector
5. Vector fields can describe the distribution of vector quantities such as forces or velocities over a region of the plane or of space. They are in common use in such areas as physics, engineering, meteorology, oceanography.
6. We can sketch a vector field by examining its defining equation to determine relative magnitudes in various locations and then drawing enough vectors to determine a pattern.
7. A vector field

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen — the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

Kokeile omaasi

⌨ Type ✎ Draw

Matematiikkaongelmasi

Ratkaise

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

gradient of x*ygradient of ln(x^2 + y^2)derivative of x^2*y wrt xgradient of x^2 - y^2

Parts of this page are adapted from Boelkins et al., Active Calculus Multivariable (CC BY-SA 4.0), OpenStax Calculus Volume 3 (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Lisää Multivariable Calculus

Functions of several variablesPartial derivatives and the gradientOptimisation in several variablesDouble and triple integrals