/*! 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 17 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 17

Sketch the following vector fields. $$\mathbf{F}=\langle x, y-x\rangle$$

Short Answer

Expert verified
In this exercise, we have sketched the vector field \(\mathbf{F} = \langle x, y-x \rangle\) by analyzing its components, creating a grid of points on the xy-plane, drawing scaled vectors at each grid point, and observing the overall pattern in the sketch. The general direction of the vectors changes across each quadrant, providing a visual representation of how the vector field behaves in different regions of the xy-plane.

Step by step solution

01

Analyzing the components of the vector field

The given vector field is \(\mathbf{F}=\langle x, y-x\rangle\). This means that at any point \((x,y)\) on the xy-plane, \(\mathbf{F}\) has two components: the first component is the x-coordinate itself, and the second component is the difference between the y-coordinate and the x-coordinate.
02

Creating a grid of points

To sketch the vector field, create a grid of points on the xy-plane. Start by choosing a range for the x and y values, for example, from -5 to 5 with a step of 1. This will give us an 11x11 grid with 121 points.
03

Drawing scaled vectors at each grid point

Now, for each point (x, y) on the grid, calculate the components of the vector field: \(f_x = x\) and \(f_y = y - x\). Since the actual vectors can be quite long, it is helpful to scale down the components by dividing them by some constant factor. For example, you could divide both components by 5 in order to reduce their magnitude by a factor of 5. Then, draw a vector from each point (x, y) on the grid with the scaled components: $$\frac{1}{5}\langle x, y-x \rangle$$.
04

Observing the pattern in the sketch

Once you have sketched the vector field through the outlined steps, pay attention to the overall pattern of the vectors. Notice how the general direction of the vectors in each quadrant changes as you move across the plane. The sketch should provide a visual representation of how the vector field behaves in each region of the xy-plane. With these steps, have now successfully sketched the vector field \(\mathbf{F}=\langle x, y-x\rangle\).

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Key Concepts

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

Vector Field Sketching
Sketching a vector field involves constructing a visual representation of the vectors at various points on the xy-plane. In the exercise, we are given the vector field \( \mathbf{F} = \langle x, y-x \rangle \).
The goal is to individually plot vectors at different coordinates to understand the overall behavior of the field. To accurately sketch, follow these steps:
  • Select a range for x and y values.
  • Place coordinates on the grid and compute vector values at these points.
  • Draw vectors using calculated values, considering a consistent scale for visibility.
Visual representation helps in understanding motion patterns in fields such as fluid dynamics or electromagnetic fields, making sketching a crucial skill.
Components of Vector Fields
Vector fields consist of vectors characterized by their components, typically denoted as \( \langle f_x, f_y \rangle \). Each vector at any location \( (x, y) \) has explicit components that determine its direction and magnitude.For our vector field \( \mathbf{F} = \langle x, y-x \rangle \), the components are:
  • First Component (f_x): x - It means the vector moves positively along the x-axis by an amount equal to x.
  • Second Component (f_y): y - x - Subtraction signifies that the y-component varies depending on both the y and x coordinates.
Understanding these components is essential in predicting how the vector field behaves at any point in space.
Grid of Points
Creating a grid of points on the xy-plane is a foundational step in sketching vector fields. To chart a vector field, you need a framework over which vectors can be systematically placed. Consider a grid with a range of values, for example, from -5 to 5 with unit increments. This forms an array of points over the plane. At each of these points, calculate the vector field components. Benefits of a Grid:
  • Establishes consistency in sketching across the xy-plane.
  • Facilitates visualization through uniform placement of vectors.
  • Allows comprehensive analysis of the vector field's behavior at multiple locations.
Gridding ensures precise and clear sketches, depicting how vectors interact across the space.
Vector Scaling
Vectors may be large and difficult to represent practically without scaling. Scaling is the process of adjusting the magnitude of vectors for more manageable sketches, ensuring that they fit within the plotting region.For the vector field \( \mathbf{F} = \langle x, y-x \rangle \), vectors can be scaled by a factor, like dividing each component by 5. This means that each calculated vector component becomes one-fifth of its original size.Advantages of Vector Scaling:
  • Enhances the clarity of vector patterns in sketches.
  • Prevents overlap of vectors, facilitating better visualization.
  • Ensures that essential details in the field design are not lost due to size distortion.
Always maintain the proportionate direction and relative length of vectors to accurately reflect the field's characteristics.

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Most popular questions from this chapter

Choosing a more convenient surface The goal is to evaluate \(A=\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S,\) where \(\mathbf{F}=\langle y z,-x z, x y\rangle\) and \(S\) is the surface of the upper half of the ellipsoid \(x^{2}+y^{2}+8 z^{2}=1\) \((z \geq 0)\) a. Evaluate a surface integral over a more convenient surface to find the value of \(A\) b. Evaluate \(A\) using a line integral.

Miscellaneous surface integrals Evaluate the following integrals using the method of your choice. Assume normal vectors point either outward or upward. $$\begin{aligned} &\iint_{S} \nabla \ln |\mathbf{r}| \cdot \mathbf{n} d S, \text { where } S \text { is the hemisphere } x^{2}+y^{2}+z^{2}=a^{2}\\\ &\text { for } z \geq 0, \text { and where } \mathbf{r}=\langle x, y, z\rangle \end{aligned}$$

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T(\text {the Laplacian of } T) .\) Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100 e^{-x^{2}+y^{2}+z^{2}}$$

Surface integrals of vector fields Find the flux of the following vector fields across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. \(\mathbf{F}=\langle-y, x, 1\rangle\) across the cylinder \(y=x^{2},\) for \(0 \leq x \leq 1\) \(0 \leq z \leq 4 ;\) normal vectors point in the general direction of the positive y-axis.

Surface area of a torus a. Show that a torus with radii \(R > r\) (see figure) may be described parametrically by \(\mathbf{r}(u, v)=\langle(R+r \cos u) \cos v\) \((R+r \cos u) \sin v, r \sin u\rangle,\) for \(0 \leq u \leq 2 \pi, 0 \leq v \leq 2 \pi\) b. Show that the surface area of the torus is \(4 \pi^{2} R r\)
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