/*! 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 15 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 15

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

Short Answer

Expert verified
Based on the step-by-step solution provided, explain if the vector field has a clockwise or counterclockwise rotation around the origin. The vector field $$\mathbf{F}=\langle y,-x\rangle$$ has a clockwise rotation around the origin, as observed in the sketch after the arrows were drawn.

Step by step solution

01

Understand the vector field

The given vector field is $$\mathbf{F}=\langle y,-x\rangle.$$ This vector field represents a 2D field in which every point (x, y) in the plane has a vector attached to it. The vector at each point has components (y, -x). This means that the x-component of the vector depends on the y-coordinate, and the y-component of the vector depends on the x-coordinate.
02

Create a grid and determine the arrows at each point

Now, create a grid of points, preferably evenly spaced, like (0,0), (1,1), (1,-1), and so on. To keep this simple, let's use a 5x5 grid with points from (-2,-2) to (+2,+2). Now, we need to determine the arrows at these grid points based on the x and y components of the vector at each point. For example, at the point (1,1) , the vector F would be \(\langle 1,-1\rangle\). Repeat this process for each grid point.
03

Draw the arrows to represent the vector field

Now that we have determined the arrows at each grid point, we proceed to draw the arrows on a graph. Remember to use the origin of the arrows at the grid points and use the determined components to draw each arrow. Make sure to use the same scale so that the graph remains consistent and easy to read. After drawing the arrows, you'll notice that the vector field $$\mathbf{F}=\langle y,-x\rangle$$ has a clockwise rotation around the origin. This sketch gives us a visual representation of how the vector field behaves across the 2D plane.

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Key Concepts

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

Understanding Vector Field Components
Vector fields are mathematical descriptions of how vectors vary in two-dimensional or three-dimensional space. Each point in the vector field has a vector with both magnitude and direction.

For the vector field \[\begin{equation}\mathbf{F}=\[\begin{equation}\mathbf{F}=\left\[\begin{equation}\mathbf{F}=\langle y,-x\rangle,\end{equation}\]\end{equation}\]\end{equation}\]the 'x-component' is given by the y-coordinate, and the 'y-component' by the negative x-coordinate of the plane. Thus, the vectors' directions and lengths change based on their position in space. Grasping this concept is key to visualize and calculate with vector fields.
Sketching Vector Fields
Sketching vector fields provides a vivid illustration of how vectors behave at various points. To sketch the vector field given by \[\begin{equation}\mathbf{F}=\langle y,-x\rangle,\end{equation}\]one starts by plotting a selection of points across the area of interest. Using a grid where vectors are placed ensures consistency and uniformity in representation.

Upon choosing a point, the vector's direction and magnitude are determined by its coordinates. For instance, at point (1, 1), the vector field would be \[\begin{equation}\mathbf{F}=\langle 1,-1\rangle.\end{equation}\]Following this, the vectors are drawn as arrows, with the arrow lengths indicative of the vectors' magnitude and the arrows' direction representing the vector's direction. Once all vectors are placed, the overall pattern or flow of the vector field emerges, which in this case, shows a clockwise rotation around the origin.
Grid Method for Vector Fields
The grid method is an effective technique for visualizing vector fields. By dividing the area into a grid, usually with equal spacing, one can more systematically and reliably map out the vector field.

When employing this method on the example field \[\begin{equation}\mathbf{F}=\langle y,-x\rangle,\end{equation}\]you would select grid points, such as along a 5x5 grid stretching from (-2,-2) to (2,2). At each grid point, the corresponding vector is determined by the coordinates of that point and drawn in place. For visual clarity, keeping the scale consistent is crucial, or else the sketch might misrepresent the vector field's behavior.

Using the grid method ensures that even individuals new to vector fields can accurately represent and analyze the variations across the field.

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Most popular questions from this chapter

The potential function for the force field due to a charge \(q\) at the origin is \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{|\mathbf{r}|},\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of a point in the field, and \(\varepsilon_{0}\) is the permittivity of free space. a. Compute the force field \(\mathbf{F}=-\nabla \varphi\) b. Show that the field is irrotational; that is, show that \(\nabla \times \mathbf{F}=\mathbf{0}\)

Streamlines and equipotential lines Assume that on \(\mathbb{R}^{2}\), the vector field \(\mathbf{F}=\langle f, g\rangle\) has a potential function \(\varphi\) such that \(f=\varphi_{x}\) and \(g=\varphi_{y},\) and it has a stream function \(\psi\) such that \(f=\psi_{y}\) and \(g=-\psi_{x}\). Show that the equipotential curves (level curves of \(\varphi\) ) and the streamlines (level curves of \(\psi\) ) are everywhere orthogonal.

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 \(k=1\) $$T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}} ; S \text { is the sphere } x^{2}+y^{2}+z^{2}=a^{2}$$

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and a parametric description of the sphere.

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla \cdot \frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}}=\frac{3-p}{|\mathbf{r}|^{p}}\)
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