/*! 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 14 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 14

Sketch the following vector fields. $$\mathbf{F}=\langle 2 x, 3 y\rangle$$

Short Answer

Expert verified
To sketch the given vector field \(\mathbf{F} = \langle 2x, 3y \rangle\), follow these steps: 1. Define the vector field as \(\mathbf{F} = \langle 2x, 3y \rangle\), which represents a vector with components \(2x\) and \(3y\) at each point \((x, y)\) in the plane. 2. Choose a set of representative points in the region of interest, such as a \(4\times 4\) grid from \(x = -2\) to \(x = 2\) and \(y = -2\) to \(y = 2\). 3. At each representative point, compute the corresponding vector \(\mathbf{F}(x,y) = \langle 2x, 3y \rangle\) and draw it with its tail at \((x,y)\) and head at \((x+2x, y+3y)\). For example, at \((1,1)\), the vector \(\mathbf{F}(1,1) = \langle 2, 3 \rangle\), and you draw it with its tail at \((1,1)\) and head at \((3,4)\). 4. Connect the head of each vector to the tail of its neighboring vectors to create a grid that represents the overall behavior of the vector field in the region of interest. Studying the patterns and trends in the magnitudes and directions of the vectors will help understand the nature of the vector field. By completing these steps, you will have sketched the vector field \(\mathbf{F}=\langle 2x, 3y \rangle\).

Step by step solution

01

Define the given vector field

The given vector field is represented by \(\mathbf{F} = \langle 2x, 3y \rangle\). This means that at each point \((x, y)\) in the plane, there is a vector with components \(2x\) and \(3y\).
02

Choose representative points

Pick a set of representative points in the region of interest (for example, a \(4\times 4\) grid from \(x = -2\) to \(x = 2\) and \(y = -2\) to \(y = 2\)) to evaluate the vector field and visualize the vectors.
03

Evaluate the vector field at each point

At each representative point \((x,y)\), compute the corresponding vector \(\mathbf{F}(x,y) = \langle 2x, 3y \rangle\). For example, at the point \((1,1)\), the vector \(\mathbf{F}(1,1) = \langle 2(1), 3(1) \rangle = \langle 2, 3 \rangle\). Draw this vector with its tail at \((1,1)\) and its head at \((1+2,1+3) = (3,4)\). Repeat this process for all the representative points on the grid.
04

Connect the vectors in a grid

To visualize the behavior of the vector field, connect the head of each vector to the tail of its neighboring vectors. This will give you a grid or arrows representing the overall behavior of the vector field in the region of interest. Observe the patterns and trends in the vectors' magnitudes and directions to understand the nature of the vector field. Finally, you will have a sketch of the vector field \(\mathbf{F}=\langle 2x, 3y \rangle\), showing its behavior in the region of interest.

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Key Concepts

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

Understanding Vectors
Vectors are fundamental in mathematics and physics. They represent quantities that have both magnitude and direction. Unlike simple numbers that can only express magnitude, vectors point towards a particular direction in space. For example, a vector could represent a force in physics that not only has a strength (magnitude) but also points in a certain direction. Vectors are commonly written in component form, like \( \langle a, b \rangle \), where \( a \) and \( b \) are numbers that indicate how much the vector extends in the x and y directions, respectively.In the exercise, the vector \( \mathbf{F} = \langle 2x, 3y \rangle \) means each vector is defined by multiplying the x position by 2 and the y position by 3. This transforms and stretches the space into a vector field.
Component Form Explained
The component form of a vector is a way to break down vectors into their qualitative aspects. It defines what impact the vector has on horizontal and vertical movements, making it easier to work with them mathematically.When a vector \( \mathbf{v} \) is described in component form \( \langle a, b \rangle \), each component \( a \) and \( b \) directly relates to its effect in the x and y directions.
In our example, \( \mathbf{F} = \langle 2x, 3y \rangle \), the vector field's x component is 2 times the x-coordinate, and the y component is 3 times the y-coordinate. Hence, the vector at each point varies as a function of its position, resulting in a dynamic field where vectors can change in length and direction at different points.
Visualization of Vector Fields
Visualizing vector fields helps understand how vectors interact over a region of space. A vector field assigns a vector to every point in this space.
To grasp a vector field visually, imagine a field full of tiny arrows; each arrow represents a vector at that point, indicating both direction and magnitude. This is useful in physics and engineering, as it helps to visualize various phenomena, such as gravitational fields or fluid flows.When visualizing the given vector field \( \mathbf{F} = \langle 2x, 3y \rangle \), each vector originates from its respective point \( (x,y) \), pointing in the direction specified by its components. This results in an increasingly large vector moving outward as you move away from the origin, suggesting an outward flow that aligns more strongly with x-direction horizontally and y-direction vertically.
Grid Method for Visualization
The grid method is a straightforward approach to visualizing a vector field by focusing on specific points in a defined rectangular region. This makes it easier to discern patterns or behavior over an area.To employ the grid method effectively:
  • Choose a grid of points evenly spread out in the region of interest, such as a \(4 \times 4\) grid.
  • Calculate the vector at each grid intersection using the vector field formula, like \( \mathbf{F}(x, y) = \langle 2x, 3y \rangle \).
  • Draw each vector starting from its grid point, illustrating both direction and magnitude.
This method provides a cleaner, more organized picture of how vector fields behave, simplifying complex scenarios into easier visual interpretations. Observing trends in how vectors connect further aids in understanding flows and patterns in the field.

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Most popular questions from this chapter

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

Heat flux The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k > 0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\) $$T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}} ; S \text { is the sphere } x^{2}+y^{2}+z^{2}=a^{2}$$

Flux across concentric spheres Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(p\) is a real number. Let \(S\) consist of the spheres \(A\) and \(B\) centered at the origin with radii \(0

Radial fields in \(\mathbb{R}^{3}\) are conservative Prove that the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number, is conservative on any region not containing the origin. For what values of \(p\) is \(\mathbf{F}\) conservative on a region that contains the origin?

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields \(\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i}\) and \(\mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}\) satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).
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