/*! 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 13 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 13

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

Short Answer

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Question: Sketch the vector field \(\mathbf{F}=\langle x, -y \rangle\) by focusing on the magnitude and direction of the vectors. Answer: To sketch the vector field \(\mathbf{F}=\langle x, -y \rangle\), create a grid of points in the \(xy\)-plane. For each point \((x, y)\), place a small arrow centered at the point that represents the vector \(\mathbf{F}(x,y)=\langle x, -y \rangle\). Focus on the direction and magnitude of the vectors, which show the arrows pointing away from the \(y\)-axis and towards the \(x\)-axis, with magnitudes increasing as you move away from the origin. The resulting sketch will effectively demonstrate the direction and magnitude of the vector field.

Step by step solution

01

Set up a grid of points in the xy-plane

To start, we will create a grid of points in the \(xy\)-plane. For this exercise, we can use a grid with points at each integer value from \(-5\) to \(5\) in both the \(x\)- and \(y\)-directions. The grid points will be labeled \((x, y)\).
02

Compute the vector at each grid point

Now, we need to compute the vector \(\mathbf{F}\) at each grid point in our coordinate system. For a given point \((x, y)\), the vector is given by \(\mathbf{F}(x,y)=\langle x, -y \rangle\).
03

Visualize the vector field using arrows

To visualize the vector field, we will place a small arrow centered at each grid point \((x, y)\), which represents the vector \(\mathbf{F}(x,y)=\langle x, -y \rangle\). The direction of the arrow should follow the direction of the vector, and its length should indicate the magnitude of the vector. To help you visualize the vector field, you can sketch the positive x-components in red, and the negative x-components in blue, using a darker color for higher magnitudes.
04

Simplify the sketch and focus on the magnitude and direction of vectors

As we add more arrows to the grid, our sketch might become cluttered. To simplify the sketch, focus on the direction and magnitude of the vectors. Note that in this vector field, the vector arrow at each point \((x,y)\) is pointing away from the \(y\)-axis and towards the \(x\)-axis. As you move away from the origin, the magnitude of the vectors increases. By completing these steps, you should be able to sketch the given vector field \(\mathbf{F}=\langle x,-y\rangle\). The final result should be a grid of arrows that effectively demonstrate the direction and magnitude of the vector field.

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Key Concepts

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

XY-Plane
The xy-plane is a two-dimensional coordinate system, crucial for visualizing vector fields like \( \mathbf{F}=\langle x,-y\rangle \). In this plane, points are defined by pairs of coordinates \( (x, y) \).

Imagine a flat surface where you can plot any point using these coordinates. For example, \( (1, 2) \) means moving one unit to the right (x-axis) and two units up (y-axis).
  • The x-axis runs horizontally, while the y-axis runs vertically.
  • Each location on this plane corresponds to a specific point with an x and y value.
Understanding the xy-plane helps you place vectors and visualize their interactions effectively. When sketching vector fields, this plane serves as the canvas for illustrating how vectors form patterns based on their components.
Magnitude of Vectors
The magnitude of a vector describes its length. For a vector \( \mathbf{F} = \langle x, -y \rangle \), the magnitude can be calculated using the formula: \[\text{Magnitude} = \sqrt{x^2 + (-y)^2} = \sqrt{x^2 + y^2}.\]\
The magnitude indicates the strength or intensity of the vector at any point.

In our example, as you move away from the origin in the xy-plane, both the x and y components increase, leading to a larger magnitude.
  • Vectors closer to the origin have a smaller magnitude, appearing shorter.
  • As you explore further from the origin, the vectors grow longer, representing increased magnitude.
Visualizing magnitude helps emphasize which vectors are more significant in the field's layout.
Direction of Vectors
The direction of a vector in the field tells us where it points. With the vector \( \mathbf{F} = \langle x, -y \rangle \), the direction is determined by its components:

- The x-component \( (x) \) points horizontally.
- The y-component \( (-y) \) points vertically but in the opposite direction of the y-axis.
  • The vector's direction is away from the y-axis and toward the x-axis.
  • Consistently, vectors point to the right (if x is positive) and downward (since y is negative).
Vectors illustrate the flux or flow within the field, showing movement patterns. Observing direction allows you to understand how the vector field behaves and predicts how surrounding forces might act upon objects within the field.

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Most popular questions from this chapter

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.

Maximum surface integral Let \(S\) be the paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0,\) where \(a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle .\) For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value?

Prove that for a real number \(p\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla \cdot \nabla\left(\frac{1}{|\mathbf{r}|^{p}}\right)=\frac{p(p-1)}{|\mathbf{r}|^{p+2}}\).

Streamlines are tangent to the vector field Assume the vector field \(\mathbf{F}=\langle f, g\rangle\) is related to the stream function \(\psi\) by \(\psi_{y}=f\) and \(\psi_{x}=-g\) on a region \(R .\) Prove that at all points of \(R,\) the vector field is tangent to the streamlines (the level curves of the stream function).

Amp猫re's Law The French physicist Andr茅-Marie Amp猫re \((1775-1836)\) discovered that an electrical current \(I\) in a wire produces a magnetic field \(\mathbf{B}\). A special case of Amp猫re's Law relates the current to the magnetic field through the equation \(\oint_{C} \mathbf{B} \cdot d \mathbf{r}=\mu I,\) where \(C\) is any closed curve through which the wire passes and \(\mu\) is a physical constant. Assume the current \(I\) is given in terms of the current density \(\mathbf{J}\) as \(I=\iint_{S} \mathbf{J} \cdot \mathbf{n} d S,\) where \(S\) is an oriented surface with \(C\) as a boundary. Use Stokes' Theorem to show that an equivalent form of Ampere's Law is \(\nabla \times \mathbf{B}=\mu \mathbf{J}\).
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