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Chapter 17: Problem 53

Determine whether the following statements are true and give an explanation or counterexample. a. The vector field \(\mathbf{F}=\left\langle 3 x^{2}, 1\right\rangle\) is a gradient field for both \(\varphi_{1}(x, y)=x^{3}+y\) and \(\varphi_{2}(x, y)=y+x^{3}+100.\) b. The vector field \(\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}\) is constant in direction and magnitude on the unit circle. c. The vector field \(\mathbf{F}=\frac{\langle y, x\rangle}{\sqrt{x^{2}+y^{2}}}\) is neither a radial field nor a rotation field.

Short Answer

Expert verified
Based on the analysis and solution, we can conclude that: a) The given vector field is a gradient field for both 蠁鈧�(x, y) and 蠁鈧�(x, y). b) The given vector field has constant direction and magnitude on the unit circle. c) The given vector field is not a radial field but is a rotation field. Therefore, both statements a and b are true, while statement c is false.

Step by step solution

01

Check if the vector field is a gradient field

To see if a vector field is a gradient field, we need to verify if the partial derivatives match the components of the vector field. We will do this for both functions 蠁鈧�(x, y) and 蠁鈧�(x, y). For 蠁鈧�(x, y)=x鲁+y: 鈭囅嗏倎 = 鈱┾垈(蠁鈧�)/鈭倄, 鈭�(蠁鈧�)/鈭倅鈱� = 鈱�3x虏, 1鈱� Since the vector field F is given as 鈱�3x虏, 1鈱�, we can see that F is a gradient field for 蠁鈧�(x, y). Now let's check for 蠁鈧�(x, y)=y+x鲁+100: 鈭囅嗏倐 = 鈱┾垈(蠁鈧�)/鈭倄, 鈭�(蠁鈧�)/鈭倅鈱� = 鈱�3x虏, 1鈱� Again, since the vector field F is 鈱�3x虏, 1鈱�, we can see that the vector field is a gradient field for 蠁鈧�(x, y) as well. So, the statement a is true.
02

Verify if the vector field has constant direction and magnitude

To check if the given vector field has constant direction and magnitude on the unit circle, we should first parameterize the unit circle and substitute x and y in the vector field. A unit circle can be parameterized as x = cos(胃) and y = sin(胃), where 0 鈮� 胃 鈮� 2蟺. So the vector field becomes: F = 鈱﹕in(胃)/sqrt(cos虏(胃)+sin虏(胃)), cos(胃)/sqrt(cos虏(胃)+sin虏(胃))鈱� = 鈱﹕in(胃), cos(胃)鈱� Since sin虏(胃) + cos虏(胃) = 1, and the direction of the vector field is determined by the values of sine and cosine, the direction and magnitude of the vector field are constant on the unit circle. So, the statement b is true.
03

Determine if the vector field is neither a radial field nor a rotation field

A radial field is a vector field that points directly away from or towards a fixed point in space (like the origin). A rotation field is a vector field that causes rotation around a fixed point in space. Let's review the vector field F=鈱﹜/sqrt(x虏+y虏), x/sqrt(x虏+y虏)鈱�. Radial field: Notice that the x component of F increases as y increases, meaning that the vector field does not point directly towards or away from the origin. So it's not a radial field. Rotation field: For a two-dimensional rotation field, the components must satisfy the condition: 鈭侳鈧�/鈭倄 - 鈭侳鈧�/鈭倅 = 0. Let's calculate the partial derivatives: 鈭侳鈧�/鈭倄 = 鈭�(y/sqrt(x虏+y虏))/鈭倄 = -(xy)/((x虏+y虏)^(3/2)) 鈭侳鈧�/鈭倅 = 鈭�(x/sqrt(x虏+y虏))/鈭倅 = -(xy)/((x虏+y虏)^(3/2)) Since 鈭侳鈧�/鈭倄 - 鈭侳鈧�/鈭倅 = 0, the vector field F is a rotation field. So, statement c is false.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gradient Field
To determine if a vector field is a gradient field, we need to see if it can be expressed as the gradient of some scalar function. A gradient field consists of vector components that match these partial derivatives. For example, given a vector field \( \mathbf{F} = \langle 3x^2, 1 \rangle \), we can check if \( \mathbf{F} \) is the gradient of two functions: \( \varphi_1(x, y) = x^3 + y \) and \( \varphi_2(x, y) = y + x^3 + 100 \).
  • Calculate partial derivatives for both functions:
    • For \( \varphi_1 \), the partial derivatives are \( \partial \varphi_1 / \partial x = 3x^2 \) and \( \partial \varphi_1 / \partial y = 1 \).
    • For \( \varphi_2 \), the partial derivatives are \( \partial \varphi_2 / \partial x = 3x^2 \) and \( \partial \varphi_2 / \partial y = 1 \).
As seen, the vector field \( \langle 3x^2, 1 \rangle \) matches these derivatives, indicating that \( \mathbf{F} \) is a gradient field for both functions.
Radial Field
A radial field is characterized by vectors that point directly away from or toward a fixed point, usually the origin. In mathematical terms, a radial field in two dimensions would have the form \( \mathbf{F} = f(r) \mathbf{r} \) where \( \mathbf{r} = \langle x, y \rangle \) and \( r \) is the distance from the point to the origin.
  • A true radial field grows or shrinks consistently from the origin, with vectors oriented in the radial direction from the center.
  • It is important to note that this radial behavior should be consistent for all points in the vector field to be classified as a radial field.
The vector field \( \mathbf{F} = \langle y / \sqrt{x^2 + y^2}, x / \sqrt{x^2 + y^2} \rangle \) increases the x-component as y increases. Hence, it does not purely radiate outwards or inwards from a fixed point, which means it is not a radial field.
Rotation Field
A rotation field differs from a radial field because it generates a rotational motion around a point. In a plane, the field that creates circular movement around a center is a rotation field. This can be checked by the condition \( \frac{\partial F_1}{\partial x} - \frac{\partial F_2}{\partial y} = 0 \). In this case, for the vector field \( \mathbf{F} = \langle y / \sqrt{x^2 + y^2}, x / \sqrt{x^2 + y^2} \rangle \):
  • Compute \( \frac{\partial F_1}{\partial x} = -\frac{xy}{(x^2 + y^2)^{3/2}} \).
  • Compute \( \frac{\partial F_2}{\partial y} = -\frac{xy}{(x^2 + y^2)^{3/2}} \).
Since \( \frac{\partial F_1}{\partial x} - \frac{\partial F_2}{\partial y} = 0 \), the vector field satisfies the condition for a rotation field. Therefore, it indeed represents a rotational movement around the point.
Unit Circle
The unit circle is a key concept in trigonometry and vector fields. It is a circle with a radius of one, centered at the origin (0, 0) in the coordinate plane. For analyzing vector fields, the unit circle simplifies understanding of directions and magnitudes.
  • The unit circle can be parameterized by \( x = \cos(\theta) \) and \( y = \sin(\theta) \), where \( 0 \leq \theta \leq 2\pi \).
  • On the unit circle \( x^2 + y^2 = 1 \) always holds true.
For the vector field \( \mathbf{F} = \langle y / \sqrt{x^2 + y^2}, x / \sqrt{x^2 + y^2} \rangle \), substituting with unit circle parameters simplifies the field to \( \mathbf{F} = \langle \sin(\theta), \cos(\theta) \rangle \). This indicates a direction following the circle itself, confirming the direction and magnitude remain constant on the unit circle.

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Most popular questions from this chapter

Alternative construction of potential functions Use the procedure in Exercise 71 to construct potential functions for the following fields. $$\quad \mathbf{F}=\langle-y,-x\rangle$$

Inverse square fields are special Let \(F\) be a radial ficld \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle .\) With \(p=3, \mathbf{F}\) is an inverse square field. a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for \(p=3\) b. Explain the observation in part (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) across the boundaries of a spherical box \(\left\\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{1} \leq \varphi \leq \varphi_{2}, \theta_{1} \leq \theta \leq \theta_{2}\right\\}\) for various values of \(p\)

Green's Second Identity Prove Green's Second Identity for scalar-valued functions \(u\) and \(v\) defined on a region \(D:\) $$\iiint_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d V=\iint_{S}(u \nabla v-v \nabla u) \cdot \mathbf{n} d S$$ (Hint: Reverse the roles of \(u\) and \(v\) in Green's First Identity.)

Zero circulation fields. For what values of \(b\) and \(c\) does the vector field \(\mathbf{F}=\langle b y, c x\rangle\) have zero circulation on the unit circle centered at the origin and oriented counterclockwise?

Radial fields and zero circulation Consider the radial vector fields \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle\) Let \(C\) be any circle in the \(x y\) -plane centered at the origin. a. Evaluate a line integral to show that the field has zero circulation on \(C\) b. For what values of \(p\) does Stokes' Theorem apply? For those values of \(p,\) use the surface integral in Stokes' Theorem to show that the field has zero circulation on \(C\).
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