How to Quickly Draw Coordinate Plane

Learning Objectives

• 6.1.1 Recognize a vector field in a plane or in space.
• half-dozen.1.two Sketch a vector field from a given equation.
• 6.1.3 Identify a conservative field and its associated potential part.

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

Examples of Vector Fields

How tin nosotros model the gravitational force exerted past multiple astronomical objects? How can we model the velocity of water particles on the surface of a river? Figure 6.two gives visual representations of such phenomena.

Figure half-dozen.2(a) shows a gravitational field exerted by two astronomical objects, such every bit a star and a planet or a planet and a moon. At any point in the figure, the vector associated with a signal 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, then it exerts a gravitational force of greater magnitude than the smaller object.

Figure half-dozen.2(b) shows the velocity of a river at points on its surface. The vector associated with a given indicate on the river's surface gives the velocity of the water at that indicate. Since the vectors to the left of the effigy are small in magnitude, the water is flowing slowly on that office 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 vi.ii (a) The gravitational field exerted past two astronomical bodies on a small object. (b) The vector velocity field of h2o on the surface of a river shows the varied speeds of water. Crimson indicates that the magnitude of the vector is greater, so the h2o flows more quickly; bluish indicates a lesser magnitude and a slower speed of water flow.

Each effigy illustrates an instance of a vector field. Intuitively, a vector field is a map of vectors. In this section, we report vector fields in ${ℝ}^2$ and ${ℝ}^3.$

Definition

A vector field $\text{F}$ in ${ℝ}^{\mathrm{ii}}$ is an assignment of a two-dimensional vector $\text{F}\left(\mathrm{ten},y\right)$ to each point $\left(\mathrm{ten},y\right)$ of a subset D of ${ℝ}^2.$ The subset D is the domain of the vector field.

A vector field F in ${ℝ}^3$ is an assignment of a three-dimensional vector $\text{F}\left(\mathrm{10},y,z\right)$ to each point $\left(x,y,z\right)$ of a subset D of ${ℝ}^3.$ The subset D is the domain of the vector field.

Vector Fields in ${ℝ}^2$

A vector field in ${ℝ}^2$ can be represented in either of ii equivalent ways. The outset manner is to use a vector with components that are two-variable functions:

$$\text{F}(x,y)=\langle P(x,y),Q(\mathrm{10},y)\rangle .$$

(6.1)

The second mode is to use the standard unit vectors:

$$\text{F}(\mathrm{ten},y)=P(\mathrm{10},y)\text{i}+Q(x,y)\text{j}.$$

(vi.ii)

A vector field is said to exist continuous if its component functions are continuous.

Instance 6.one

Finding a Vector Associated with a Given Point

Allow $\text{F}(\mathrm{ten},y)=(2y^{\mathrm{two}}+\mathrm{ten}-4)\text{i}+\text{cos}(\mathrm{10})\text{j}$ 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 $\left(0,\mathrm{-ane}\right)?$

Checkpoint 6.ane

Let $\text{One thousand}\left(\mathrm{10},y\right)=x^{2}y\text{i}-\left(\mathrm{10}+y\right)\text{j}$ exist a vector field in ${ℝ}^{\mathrm{ii}}.$ What vector is associated with the point $\left(\mathrm{-2},\mathrm{three}\right)?$

Cartoon a Vector Field

We tin 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 considering the domain of a vector field is in ${ℝ}^2,$ every bit is the range. Therefore the "graph" of a vector field in ${ℝ}^2$ lives in 4-dimensional space. Since we cannot represent four-dimensional infinite visually, we instead draw vector fields in ${ℝ}^2$ in a aeroplane itself. To practise this, depict the vector associated with a given signal at the point in a plane. For example, suppose the vector associated with point $\left(4,\mathrm{-one}\right)$ is $\langle 3,1\rangle .$ Then, we would draw vector $\langle 3,\mathrm{ane}\rangle$ at point $\left(4,\mathrm{-1}\right).$

Nosotros should plot plenty vectors to run into the general shape, but not so many that the sketch becomes a jumbled mess. If we were to plot the epitome vector at each point in the region, information technology would fill up 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 sure 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 straight toward or direct away from the origin. Furthermore, the magnitude of whatever vector depends just on its altitude from the origin. In a radial field, the vector located at point $\left(x,y\right)$ is perpendicular to the circumvolve centered at the origin that contains indicate $\left(x,y\right),$ and all other vectors on this circumvolve take the same magnitude.

Instance 6.2

Drawing a Radial Vector Field

Sketch the vector field $\text{F}(\mathrm{10},y)=\frac{x}{2}\text{i}+\frac{y}{2}\text{j}.$

Checkpoint 6.two

Describe the radial field $\text{F}(\mathrm{10},y)=-\frac{x}{3}\text{i}-\frac{y}{\mathrm{three}}\text{j}.$

In dissimilarity to radial fields, in a rotational field, the vector at bespeak $\left(x,y\right)$ is tangent (not perpendicular) to a circle with radius $r=\sqrt{x^{\mathrm{ii}}+y^2}.$ In a standard rotational field, all vectors point either in a clockwise direction or in a counterclockwise management, and the magnitude of a vector depends but on its altitude from the origin. Both of the following examples are clockwise rotational fields, and we see from their visual representations that the vectors appear to rotate effectually the origin.

Instance six.iii

Affiliate Opener: Drawing a Rotational Vector Field

Effigy 6.4 (credit: modification of piece of work past NASA)

Sketch the vector field $\text{F}\left(\mathrm{10},y\right)=\langle y,\text{−}\mathrm{ten}\rangle .$

Analysis

Note that vector $\text{F}\left(a,b\right)=\langle b,\text{−}a\rangle$ points clockwise and is perpendicular to radial vector $\langle a,b\rangle .$ (We can verify this assertion past computing the dot product of the two vectors: $\langle a,b\rangle ·\langle \text{−}b,a\rangle =\text{−}ab+ab=0.)$ Furthermore, vector $\langle b,\text{−}a\rangle$ has length $r=\sqrt{a^2+b^{\mathrm{two}}}.$ Thus, we have a complete clarification of this rotational vector field: the vector associated with betoken $\left(a,b\right)$ is the vector with length r tangent to the circumvolve with radius r, and it points in the clockwise management.

Sketches such as that in Figure half dozen.half dozen are often used to clarify major storm systems, including hurricanes and cyclones. In the northern hemisphere, storms rotate counterclockwise; in the southern hemisphere, storms rotate clockwise. (This is an consequence caused by Earth's rotation near its axis and is called the Coriolis Effect.)

Example 6.4

Sketching a Vector Field

Sketch vector field $\text{F}(\mathrm{ten},y)=\frac{y}{x^2+y^{\mathrm{ii}}}\text{i}-\frac{\mathrm{ten}}{x^{\mathrm{two}}+y^{\mathrm{two}}}\text{j}.$

Checkpoint 6.iii

Sketch vector field $\text{F}\left(\mathrm{ten},y\right)=\langle \mathrm{-2}y,2x\rangle .$ Is the vector field radial, rotational, or neither?

Instance 6.5

Velocity Field of a Fluid

Suppose that $\text{v}\left(x,y\right)=-\frac{\mathrm{two}y}{{\mathrm{10}}^{\mathrm{ii}}+y^2}\text{i}+\frac{2x}{{\mathrm{ten}}^{\mathrm{ii}}+y^2}\text{j}$ is the velocity field of a fluid. How fast is the fluid moving at point $\left(1,\mathrm{-1}\right)?$ (Presume the units of speed are meters per second.)

Checkpoint 6.iv

Vector field $v(\mathrm{ten},y)=\langle 4\left|\mathrm{10}\right|,1\rangle$ models the velocity of water on the surface of a river. What is the speed of the water at point $\left(2,3\right)?$ Use meters per 2d as the units.

We have examined vector fields that incorporate vectors of various magnitudes, but just as we have unit vectors, we tin also have a unit vector field. A vector field F is a unit vector field if the magnitude of each vector in the field is 1. In a unit of measurement vector field, the only relevant data is the direction of each vector.

Example 6.6

A Unit Vector Field

Show that vector field $\text{F}\left(x,y\right)=\langle \frac{y}{\sqrt{x^2+y^2}},-\frac{x}{\sqrt{x^2+y^{\mathrm{ii}}}}\rangle$ is a unit of measurement vector field.

Checkpoint half dozen.5

Is vector field $\text{F}\left(\mathrm{ten},y\right)=\langle \text{−}y,x\rangle$ a unit of measurement vector field?

Why are unit vector fields important? Suppose we are studying the menses of a fluid, and nosotros care only virtually the direction in which the fluid is flowing at a given point. In this case, the speed of the fluid (which is the magnitude of the corresponding velocity vector) is irrelevant, because all we intendance near is the direction of each vector. Therefore, the unit vector field associated with velocity is the field we would study.

If $\text{F}=\langle P,Q,R\rangle$ is a vector field, then the corresponding unit vector field is $\langle \frac{P}{\left|\left|\text{F}\right|\right|},\frac{Q}{\left|\left|\text{F}\right|\right|},\frac{R}{\left|\left|\text{F}\right|\right|}\rangle .$ Notice that if $\text{F}\left(x,y\right)=\langle y,\text{−}x\rangle$ is the vector field from Example 6.3, then the magnitude of F is $\sqrt{x^2+y^2},$ and therefore the corresponding unit of measurement vector field is the field Grand from the previous example.

If F is a vector field, then the procedure of dividing F past its magnitude to class unit vector field $\text{F}\text{/}\left|\left|\text{F}\right|\right|$ is called normalizing the field F.

Vector Fields in ${ℝ}^3$

We have seen several examples of vector fields in ${ℝ}^{\mathrm{ii}};$ let'due south now turn our attention to vector fields in ${ℝ}^3.$ These vector fields can be used to model gravitational or electromagnetic fields, and they can as well be used to model fluid flow or heat flow in three dimensions. A two-dimensional vector field can really simply model the movement of water on a ii-dimensional piece of a river (such as the river's surface). Since a river flows through 3 spatial dimensions, to model the catamenia of the unabridged depth of the river, we need a vector field in iii dimensions.

The actress dimension of a three-dimensional field tin make vector fields in ${ℝ}^3$ more hard to visualize, merely the idea is the aforementioned. To visualize a vector field in ${ℝ}^3,$ plot plenty vectors to show the overall shape. We can apply a like method to visualizing a vector field in ${ℝ}^2$ by choosing points in each octant.

Just as with vector fields in ${ℝ}^{\mathrm{two}},$ we can stand for vector fields in ${ℝ}^3$ with component functions. We only demand an actress component part for the extra dimension. We write either

$$\text{F}\left(x,y,z\right)=\langle P\left(\mathrm{10},y,z\right),Q\left(x,y,z\right),R\left(x,y,z\right)\rangle$$

(6.iii)

or

$$\text{F}\left(x,y,z\right)=P\left(\mathrm{ten},y,z\right)\text{i}+Q\left(x,y,z\right)\text{j}+R\left(x,y,z\right)\text{grand}.$$

(half-dozen.four)

Example six.7

Sketching a Vector Field in Iii Dimensions

Describe vector field $\text{F}\left(\mathrm{ten},y,z\right)=\langle \mathrm{ane},\mathrm{one},z\rangle .$

Checkpoint vi.6

Sketch vector field $\text{M}\left(x,y,z\right)=\langle \mathrm{two},\frac{z}{\mathrm{two}},\mathrm{ane}\rangle .$

In the next instance, nosotros explore ane of the classic cases of a 3-dimensional vector field: a gravitational field.

Example 6.viii

Describing a Gravitational Vector Field

Newton's law of gravitation states that $\text{F}=\text{−}G\frac{{\mathrm{1000}}_{\mathrm{ane}}{k}_2}{r^{\mathrm{ii}}}\widehat{r},$ where Chiliad is the universal gravitational constant. It describes the gravitational field exerted by an object (object 1) of mass $m_{\mathrm{one}}$ located at the origin on some other object (object two) of mass $m_{\mathrm{two}}$ located at point $\left(\mathrm{ten},y,z\right).$ Field F denotes the gravitational force that object 1 exerts on object 2, r is the distance between the two objects, and $\widehat{r}$ 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 forcefulness of object 1 is attractive. Sketch the vector field associated with this equation.

Checkpoint 6.7

The mass of asteroid i is 750,000 kg and the mass of asteroid two is 130,000 kg. Assume asteroid 1 is located at the origin, and asteroid 2 is located at $\left(15,\mathrm{-5},\mathrm{ten}\right),$ measured in units of 10 to the eighth power kilometers. Given that the universal gravitational constant is $G=\mathrm{half\; dozen.67384}\times {10}^{\mathrm{-eleven}}{\text{m}}^3{\text{kg}}^{\mathrm{-1}}{\text{s}}^{\mathrm{-2}},$ notice the gravitational forcefulness vector that asteroid one exerts on asteroid two.

Gradient Fields

In this department, nosotros study a special kind of vector field called a gradient field or a conservative field. These vector fields are extremely important in physics considering they can be used to model physical systems in which energy is conserved. Gravitational fields and electric fields associated with a static accuse are examples of gradient fields.

Call up that if $f$ is a (scalar) function of x and y, and then the gradient of $f$ is

$$\text{grad}f=\text{∇}f=f_x(x,y)\text{i}+f_y(x,y)\text{j}.$$

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

$$\text{grad}f=\text{∇}f=f_{\mathrm{ten}}(\mathrm{ten},y,z)\text{i}+f_y(x,y,z)\text{j}+f_z(x,y,z)\text{m}.$$

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

A gradient field is a vector field that tin exist written equally the gradient of a office, and nosotros have the following definition.

Definition

A vector field $\text{F}$ in ${ℝ}^2$ or in ${ℝ}^3$ is a gradient field if there exists a scalar office $f$ such that $\text{∇}f=\text{F}.$

Example 6.ix

Sketching a Gradient Vector Field

Use technology to plot the gradient vector field of $f\left(\mathrm{10},y\right)={\mathrm{ten}}^2{y}^{\mathrm{two}}.$

Checkpoint 6.8

Use technology to plot the gradient vector field of $f\left(x,y\right)=\text{sin}\mathrm{10}\text{cos}y.$

Consider the function $f\left(x,y\right)=x^2{y}^2$ from Example 6.ix. Effigy 6.11 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 point the graph is steep, and the magnitude of the gradient vector is the largest value of the directional derivative. Therefore, you tin run across the local steepness of a graph past investigating the corresponding function's gradient field.

Figure half dozen.10 The gradient field of $f\left(\mathrm{10},y\right)={\mathrm{10}}^2{y}^2$ and several level curves of $f.$ Notice that every bit the level curves become closer together, the magnitude of the gradient vectors increases.

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

Example 6.10

Verifying a Potential Function

Is $f\left(x,y,z\right)=x^{\mathrm{ii}}yz-\text{sin}\left(xy\right)$ a potential role for vector field

$$\text{F}(x,y,z)=\langle \mathrm{two}xyz-y\text{cos}(xy),x^{2}z-\mathrm{ten}\text{cos}(xy),x^{\mathrm{ii}}y\rangle ?$$

Checkpoint 6.nine

Is $f(\mathrm{ten},y,z)={\mathrm{10}}^2\text{cos}(yz)+y^2{z}^2$ a potential function for $\text{F}(\mathrm{10},y,z)=\langle 2x\text{cos}(yz),\text{−}{\mathrm{10}}^{2}z\text{sin}(yz)+2y{z}^2,y^2\rangle ?$

Example 6.11

Verifying a Potential Role

The velocity of a fluid is modeled by field $\text{5}\left(x,y\right)=\langle xy,\frac{x^2}{\mathrm{ii}}-y\rangle .$ Verify that $f\left(x,y\right)=\frac{x^{2}y}{\mathrm{two}}-\frac{y^2}{2}$ is a potential function for v.

Checkpoint six.10

Verify that $f\left(x,y\right)=x^3{y}^2+x$ is a potential office for velocity field $\mathbf{\text{v}}(x,y)=⟨\mathrm{two}x{y}^2+\mathrm{one},\mathrm{two}{x}^{2}y⟩.$

If F is a bourgeois vector field, then there is at least one potential function $f$ such that $\text{∇}f=\text{F}.$ But, could at that place be more than i potential part? If then, is there any relationship betwixt two potential functions for the same vector field? Earlier answering these questions, permit's recall some facts from unmarried-variable calculus to guide our intuition. Remember that if $g\left(x\right)$ is an integrable function, then g has infinitely many antiderivatives. Furthermore, if F and Thousand are both antiderivatives of k, then F and G differ only past a constant. That is, at that place is some number C such that $F\left(x\right)=K\left(x\right)+C.$

At present let $\mathbf{F}$ be a bourgeois vector field and permit $f$ and g exist potential functions for $\mathbf{F}$ . Since the slope is like a derivative, $\mathbf{F}$ being conservative means that $\mathbf{F}$ 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 $f\left(\mathrm{10}\right)=g\left(x\right)+C.$ The next theorem says that this is indeed the instance.

To state the next theorem with precision, we need to presume the domain of the vector field is connected and open. To be continued means if $P_{\mathrm{ane}}$ and $P_{\mathrm{two}}$ are whatsoever two points in the domain, then you lot tin can walk from $P_1$ to $P_2$ along a path that stays entirely inside the domain.

Theorem half-dozen.1

Uniqueness of Potential Functions

Let F be a conservative vector field on an open up and connected domain and let $f$ and chiliad be functions such that $\text{∇}f=\text{F}$ and $\text{∇}\mathrm{one\; thousand}=\text{F}.$ And then, in that location is a constant C such that $f=g+C.$

Proof

Since $f$ and g are both potential functions for F, and so $\text{∇}\left(f-\mathrm{thousand}\right)=\text{∇}f-\text{∇}g=\text{F}-\text{F}=0.$ Let $h=f-g,$ then we have $\text{∇}h=0.$ Nosotros would similar to bear witness that h is a constant role.

Assume h is a function of x and y (the logic of this proof extends to whatever number of contained variables). Since $\text{∇}h=0,$ we have $h_x=0$ and $h_y=0.$ The expression $h_x=0$ implies that h is a abiding function with respect to x—that is, $h\left(x,y\right)={\mathrm{chiliad}}_1\left(y\right)$ for some office yard₁ . Similarly, $h_y=0$ implies $h\left(x,y\right)=k_{\mathrm{ii}}\left(x\right)$ for some role k₂ . Therefore, function h depends but on y and also depends only on x. Thus, $h\left(x,y\right)=C$ for some constant C on the continued domain of F. Notation that nosotros really do need connectedness at this point; if the domain of F came in 2 dissever pieces, so k could exist a abiding C₁ on one piece simply could be a dissimilar constant C₂ on the other piece. Since $f-g=h=C,$ nosotros accept that $f=g+C,$ as desired.

□

Conservative vector fields as well accept a special holding called the cantankerous-partial belongings . This property helps test whether a given vector field is conservative.

Theorem half-dozen.two

The Cross-Partial Property of Bourgeois 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-fractional derivatives on the domain of F.

If $\text{F}(x,y)=\langle P(\mathrm{ten},y),Q(\mathrm{10},y)\rangle$ is a conservative vector field in ${ℝ}^2,$ then $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}.$ If $\text{F}\left(x,y,z\right)=\langle P\left(x,y,z\right),Q\left(x,y,z\right),R\left(x,y,z\right)\rangle$ is a conservative vector field in ${ℝ}^3,$ then

$$\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x},\frac{\partial Q}{\partial z}=\frac{\partial R}{\partial y},\text{and}\frac{\partial R}{\partial x}=\frac{\partial P}{\partial z}.$$

Proof

Since F is conservative, there is a function $f(\mathrm{10},y)$ such that $\text{∇}f=\text{F}.$ Therefore, by the definition of the gradient, $f_x=P$ and $f_y=Q.$ Past Clairaut's theorem, $f_{xy}=f_{y\mathrm{10}},$ But, $f_{\mathrm{10}y}=P_y$ and $f_{y\mathrm{ten}}=Q_{\mathrm{10}},$ and thus $P_y=Q_{\mathrm{10}}.$

□

Clairaut'southward theorem gives a fast proof of the cross-partial holding of bourgeois vector fields in ${ℝ}^{\mathrm{iii}},$ simply as it did for vector fields in ${ℝ}^2.$

The Cross-Partial Property of Conservative Vector Fields shows that most vector fields are non conservative. The cantankerous-partial property is difficult to satisfy in full general, then most vector fields won't have equal cantankerous-partials.

Example half-dozen.12

Showing a Vector Field Is Not Bourgeois

Show that rotational vector field $\text{F}\left(x,y\right)=\langle y,\text{−}\mathrm{ten}\rangle$ is non conservative.

Checkpoint half dozen.11

Bear witness that vector field $\text{F}\left(x,y\right)x=y\text{i}-x^{2}y\text{j}$ is not bourgeois.

Example vi.13

Showing a Vector Field Is Not Conservative

Is vector field $\text{F}\left(\mathrm{10},y,z\right)=\langle 7,\mathrm{-ii},x^3\rangle$ conservative?

Checkpoint 6.12

Is vector field $G\left(x,y,z\right)=\langle y,\mathrm{ten},\mathrm{10}yz\rangle$ bourgeois?

We conclude this section with a word of alarm: The Cross-Fractional Property of Bourgeois Vector Fields says that if F is conservative, and then F has the cantankerous-fractional property. The theorem does non say that, if F has the cross-partial property, so F is conservative (the converse of an implication is not logically equivalent to the original implication). In other words, The Cantankerous-Partial Property of Bourgeois Vector Fields tin but help make up one's mind that a field is non bourgeois; it does not let you conclude that a vector field is conservative. For instance, consider vector field $\text{F}(\mathrm{ten},y)=\langle {\mathrm{ten}}^{2}y,\frac{{\mathrm{10}}^3}{3}\rangle .$ This field has the cross-partial property, so it is natural to endeavor to use The Cross-Partial Property of Bourgeois Vector Fields to conclude this vector field is conservative. However, this is a misapplication of the theorem. We learn later how to conclude that F is bourgeois.

Section half-dozen.1 Exercises

1.

The domain of vector field $\text{F}=\text{F}\left(x,y\right)$ is a set of points $\left(x,y\right)$ in a plane, and the range of F is a set up of what in the airplane?

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

2 .

Vector field $\text{F}=\langle \mathrm{three}{x}^{\mathrm{ii}},1\rangle$ is a gradient field for both ${\varphi }_1\left(x,y\right)=x^3+y$ and ${\varphi }_{\mathrm{two}}\left(x,y\right)=y+x^3+100.$

three.

Vector field $\text{F}=\frac{\langle y,\mathrm{10}\rangle }{\sqrt{x^{\mathrm{ii}}+y^{\mathrm{two}}}}$ is constant in direction and magnitude on a unit circle.

4 .

Vector field $\text{F}=\frac{\langle y,x\rangle }{\sqrt{x^2+y^2}}$ is neither a radial field nor a rotation.

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

5.

[T] $\text{F}(x,y)=\mathrm{10}\text{i}+y\text{j}$

vi .

[T] $\text{F}(x,y)=\text{−}y\text{i}+\mathrm{ten}\text{j}$

7.

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

8 .

[T] $\text{F}(\mathrm{ten},y)=\text{i}+\text{j}$

9.

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

ten .

[T] $\text{F}(x,y)=\mathrm{iii}\text{i}+x\text{j}$

11.

[T] $\text{F}(\mathrm{10},y)=y\text{i}+\text{sin}x\text{j}$

12 .

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

13.

[T] $\text{F}(x,y,z)=2\mathrm{10}\text{i}-\mathrm{ii}y\text{j}-\mathrm{ii}z\text{k}$

14 .

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

For the following exercises, detect the gradient vector field of each function $f.$

fifteen.

$f(\mathrm{ten},y)=x\text{sin}y+\text{cos}y$

xvi .

$f(x,y,z)=z{e}^{\text{−}\mathrm{ten}y}$

17.

$f(x,y,z)={\mathrm{ten}}^{\mathrm{two}}y+\mathrm{10}y+y^{\mathrm{ii}}z$

18 .

$f(x,y)=x^2\text{sin}(\mathrm{five}y)$

19.

$f(x,y)=\text{ln}\left(\mathrm{i}+x^2+2y^2\right)$

20 .

$f(\mathrm{ten},y,z)=x\text{cos}\left(\frac{y}{z}\right)$

21.

What is vector field $\text{F}\left(x,y\right)$ with a value at $\left(x,y\right)$ that is of unit length and points toward $\left(\mathrm{i},0\right)?$

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

22 .

All vectors are parallel to the x-axis and all vectors on a vertical line take the aforementioned magnitude.

23.

All vectors signal toward the origin and have constant length.

24 .

All vectors are of unit length and are perpendicular to the position vector at that point.

25.

Give a formula $\text{F}(\mathrm{10},y)=\mathrm{1000}(x,y)\text{i}+N(x,y)\text{j}$ for the vector field in a aeroplane that has the properties that $\text{F}=0$ at $\left(0,0\right)$ and that at any other signal $\left(a,b\right),$ F is tangent to circumvolve $x^{\mathrm{ii}}+y^2=a^{\mathrm{ii}}+b^{\mathrm{ii}}$ and points in the clockwise direction with magnitude $\left|\text{F}\right|=\sqrt{a^2+b^{\mathrm{two}}}.$

26 .

Is vector field $\text{F}(\mathrm{10},y)=\left(P(\mathrm{ten},y),Q(x,y)\right)=\left(\text{sin}x+y\right)\text{i}+\left(\text{cos}y+x\right)\text{j}$ a gradient field?

27.

Find a formula for vector field $\text{F}(x,y)=\text{One thousand}(x,y)\text{i}+N(x,y)\text{j}$ given the fact that for all points $(\mathrm{10},y),$ F points toward the origin and $\left|\text{F}\right|=\frac{10}{x^2+y^2}.$

For the post-obit exercises, assume that an electric field in the xy-aeroplane acquired by an space line of charge along the x-axis is a gradient field with potential function $V\left(x,y\right)=c\text{ln}\left(\frac{r_0}{\sqrt{{\mathrm{10}}^2+y^2}}\right),$ where $c>0$ is a constant and $r_0$ is a reference distance at which the potential is assumed to be zero.

28 .

Find the components of the electric field in the x- and y-directions, where $\text{E}\left(x,y\right)=\text{−}\text{∇}V\left(x,y\right).$

29.

Testify that the electric field at a indicate in the xy-plane is directed outward from the origin and has magnitude $\left|\text{Eastward}\right|=\frac{c}{r},$ where $r=\sqrt{x^{\mathrm{two}}+y^2}.$

A catamenia line (or streamline) of a vector field $\text{F}$ is a bend $\text{r}\left(t\right)$ such that $d\text{r}\text{/}dt=\text{F}\left(\text{r}\left(t\right)\right).$ If $\text{F}$ represents the velocity field of a moving particle, and then the flow lines are paths taken by the particle. Therefore, period lines are tangent to the vector field. For the following exercises, prove that the given bend $\text{c}\left(t\right)$ is a flow line of the given velocity vector field $\text{F}\left(\mathrm{ten},y,z\right).$

30 .

$\text{c}\left(t\right)=\left(e^{2t},\text{ln}\left|t\right|,\frac{1}{t}\right),t\ne 0;\text{F}\left(x,y,z\right)=\langle 2x,z,\text{−}{z}^2\rangle$

31.

$\text{c}\left(t\right)=\left(\text{sin}t,\text{cos}t,e^t\right);\text{F}\left(\mathrm{ten},y,z\right)=\langle y,\text{−}x,z\rangle$

For the post-obit exercises, permit $\text{F}=x\text{i}+y\text{j},$ $\text{G}=\text{−}y\text{i}+x\text{j},$ and $\text{H}=\mathrm{ten}\text{i}-y\text{j}.$ Lucifer F, G, and H with their graphs.

32 .

34 .

For the following exercises, let $\text{F}=\mathrm{ten}\text{i}+y\text{j},$ $\text{G}=\text{−}y\text{i}+x\text{j},$ and $\text{H}=x\text{i}–y\text{j}.$ Match the vector fields with their graphs in $\left(\text{I}\right)-\left(\text{Iv}\right).$

1. $\text{F}+\text{G}$
2. $\text{F}+\text{H}$
3. $\text{G}+\text{H}$
4. $\text{−}\text{F}+\text{G}$

36 .

38 .

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Source: https://openstax.org/books/calculus-volume-3/pages/6-1-vector-fields

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