/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 2 Sketch the vector field \(\mathb... [FREE SOLUTION] | 91影视

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Chapter 14: Problem 2

Sketch the vector field \(\mathbf{F}=\langle x, y\rangle.\)

Short Answer

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Based on the step-by-step solution above, describe how to sketch the vector field \(\mathbf{F}=\langle x, y\rangle\). To sketch the vector field \(\mathbf{F}=\langle x, y\rangle\), follow these steps: 1. Identify the vector field components: In this case, the x-component is \(x\) and the y-component is \(y\). 2. Choose points on the coordinate plane: Select several points like (-2,2), (-1,1), (0,0), (1,1), and (2,2) which have integer values that are easy to label. 3. Sketch vectors at chosen points: For each point (x, y), draw an arrow with length and direction determined by the x-component and y-component. For example, at the point (1, 1), the vector is \(\langle1, 1\rangle\), so draw an arrow pointing up and to the right. 4. Complete the vector field sketch: Continue choosing points and sketching vectors across the entire 2D plane, keeping in mind the symmetry of the vector field, to produce a clear visualization of the vector field \(\mathbf{F}=\langle x, y\rangle\).

Step by step solution

01

Identify the vector field components

The given vector field is \(\mathbf{F}=\langle x, y\rangle\). So, the x-component is simply \(x\) and the y-component is \(y\).
02

Choose points on the coordinate plane

Choose several points on the coordinate plane (preferably on a grid). You may want to choose points where the integer values of x and y are easy to label, such as near the center of the plane. For instance, you can choose the points (-2,2), (-1,1), (0,0), (1,1), and (2,2).
03

Sketch vectors at chosen points

For each point (x, y), draw an arrow starting at that point with a length and direction determined by the x-component and y-component of the vector field, which for this problem, are the same as the x and y coordinates themselves. 1. At the point (-2, 2), the vector is \(\langle-2, 2\rangle\). Draw an arrow pointing up and to the left. 2. At the point (-1, 1), the vector is \(\langle-1, 1\rangle\). Draw an arrow pointing up and to the left, but shorter than the first arrow. 3. At the point (0, 0), the vector is \(\langle0, 0\rangle\). This represents no movement or a zero vector, so there's no need to draw an arrow at this point. 4. At the point (1, 1), the vector is \(\langle1, 1\rangle\). Draw an arrow pointing up and to the right, the same length as the arrow at the point (-1, 1). 5. At the point (2, 2), the vector is \(\langle2, 2\rangle\). Draw an arrow pointing up and to the right, the same length as the arrow at the point (-2, 2).
04

Complete the vector field sketch

Continue the process of choosing points and sketching vectors for a more complete representation of the vector field. Keep in mind the symmetry of the vector field and try to cover the entire 2D plane to produce a clear visualization of the vector field \(\mathbf{F}=\langle x, y\rangle\).

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Most popular questions from this chapter

Let \(R\) be a region in a plane that has a unit normal vector \(\mathbf{n}=\langle a, b, c\rangle\) and boundary \(C .\) Let \(\mathbf{F}=\langle b z, c x, a y\rangle\) a. Show that \(\nabla \times \mathbf{F}=\mathbf{n}\) b. Use Stokes' Theorem to show that $$\operatorname{area} \text { of } R=\oint_{C} \mathbf{F} \cdot d \mathbf{r}$$ c. Consider the curve \(C\) given by \(\mathbf{r}=\langle 5 \sin t, 13 \cos t, 12 \sin t\rangle\) for \(0 \leq t \leq 2 \pi .\) Prove that \(C\) lies in a plane by showing that \(\mathbf{r} \times \mathbf{r}^{\prime}\) is constant for all \(t\) d. Use part (b) to find the area of the region enclosed by \(C\) in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of \(C\).)

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{G} \cdot \nabla) \mathbf{F}+(\mathbf{F} \cdot \nabla) \mathbf{G}+\mathbf{G} \times(\nabla \times \mathbf{F})+\mathbf{F} \times(\nabla \times \mathbf{G})$$

Consider the vector field \(\mathbf{F}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k}\) a. Show that \(\nabla \times \mathbf{F}=\mathbf{0}\) b. Show that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is not zero on a circle \(C\) in the \(x y\) -plane enclosing the origin. c. Explain why Stokes' Theorem does not apply in this case.

The potential function for the gravitational force field due to a mass \(M\) at the origin acting on a mass \(m\) is \(\varphi=G M m /|\mathbf{r}|,\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of the mass \(m\) and \(G\) is the gravitational constant. a. Compute the gravitational force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is, \(\nabla \times \mathbf{F}=\mathbf{0}\).

Let \(\mathbf{F}\) be a radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) 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 \(p\)
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