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Chapter 17: Problem 9

Sketch the following vector fields. $$\mathbf{F}=\langle-1,1\rangle$$

Short Answer

Expert verified
Question: Sketch the given vector field \(\mathbf{F}=\langle -1, 1 \rangle\), and provide information about its structure, behavior, magnitude, and direction. Answer: The given vector field \(\mathbf{F}=\langle -1, 1 \rangle\) has a constant magnitude of \(\sqrt{2}\) and direction of \(-45^\circ\). When sketched on a coordinate plane, the arrows representing this vector field have the same length and direction, showing a consistent behavior throughout the plane. Since the arrows point diagonally down towards the left, the vector field displays a downward leftward flow.

Step by step solution

01

Identify the vector field components.

Since the vector field is given as \(\mathbf{F}=\langle -1, 1 \rangle\), the components for the field are \(F_x = -1\) and \(F_y = 1\).
02

Define the magnitude and direction of vector function

The magnitude of \(\mathbf{F}\) can be found using the formula \(|\mathbf{F}| = \sqrt{F_x^2 + F_y^2}\). In this case, we have \(|\mathbf{F}| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\). The direction (angle \(\theta\)) of the vector field can be found using the formula \(\tan^{-1}(\frac{F_y}{F_x})\). In this case, we have \(\theta = \tan^{-1}(\frac{1}{-1}) = \tan^{-1}(-1) = -45^\circ\) (please note, if necessary, adjust the angle according to the quadrant where the angle lies).
03

Pick some sample points to sketch the vector field

To sketch the vector field, we pick a few sample points on the coordinate plane and draw arrows conforming to the given magnitude and direction. These arrows represent the vector field at the chosen locations. Some of the potential points are: \((0,0), (1,0), (0,1), (-1,0),(0,-1), (1, 1), (-1, 1), (1, -1)\), and \((-1, -1)\).
04

Sketch arrows to represent \(\mathbf{F}\) at the sample points.

Plot the chosen sample points and sketch the arrows representing the vector field. Keep in mind that the arrows have the same magnitude and direction throughout the plane, that is \(\sqrt{2}\) and \(-45^\circ\) respectively. The resulting sketch will provide a visual representation of the behavior of the given vector field \(\mathbf{F}=\langle -1, 1 \rangle\).

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Key Concepts

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

Magnitude
When working with vector fields, understanding the magnitude is essential. The magnitude is essentially the length of the vector, regardless of its direction.

To find the magnitude of a vector such as \(\mathbf{F}=\langle -1, 1 \rangle\), use the formula:
  • Magnitude, \( |\mathbf{F}| = \sqrt{F_x^2 + F_y^2} \).
For \(\mathbf{F}\), this becomes:
  • \(|\mathbf{F}| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}\).
This tells us the length of the vector, \(\sqrt{2}\), which is constant across the entire plane. Knowing the magnitude helps us understand how strong or significant the vector is at any point.
Direction
The direction of a vector tells us which way it points in the coordinate plane. This is typically measured as an angle with respect to the positive x-axis.
  • Use the formula: \( \theta = \tan^{-1}(\frac{F_y}{F_x}) \).
For the vector field \( \mathbf{F}=\langle -1, 1 \rangle \), it calculates as:
  • \( \theta = \tan^{-1}(\frac{1}{-1}) = \tan^{-1}(-1) = -45^\circ \).
However, bear in mind that the angle should be adjusted based on the quadrant where the vector lies. For \(\mathbf{F}\), this means correctly plotting it in the second quadrant.
Coordinate Plane
The coordinate plane is a two-dimensional space where vectors can be visualized and analyzed. By picking sample points on the plane, we see how the vector field behaves over a region.

In this exercise, the vector \(\mathbf{F}=\langle -1, 1 \rangle\) is plotted using sample points:
  • Examples include \( (0,0), (1,0), (0,1), (-1,0),(0,-1), (1, 1), (-1, 1), (1, -1), (-1, -1) \).
Vectors at these points will have the same magnitude and direction, showcasing a repetitive pattern across the plane. This forms a clear visual representation of the vector field's behavior.
Vector Components
Vector components are the parts that make up the vector along the axes of the coordinate plane. Each vector is split into horizontal (x-axis) and vertical (y-axis) components.

For a vector like \( \mathbf{F}=\langle -1, 1 \rangle \):
  • The x-component (\(F_x\)) is -1.
  • The y-component (\(F_y\)) is 1.
These components help in calculating both magnitude and direction. By breaking vectors into components, it's easier to perform calculations and understand how vectors operate in different directions.

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Most popular questions from this chapter

Cone and sphere The cone \(z^{2}=x^{2}+y^{2},\) for \(z \geq 0,\) cuts the sphere \(x^{2}+y^{2}+z^{2}=16\) along a curve \(C\) a. Find the surface area of the sphere below \(C,\) for \(z \geq 0\) b. Find the surface area of the sphere above \(C\) c. Find the surface area of the cone below \(C\), for \(z \geq 0\)

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

A scalar-valued function \(\varphi\) is harmonic on a region \(D\) if \(\nabla^{2} \varphi=\nabla \cdot \nabla \varphi=0\) at all points of \(D\). Show that if \(u\) is harmonic on a region \(D\) enclosed by a surface \(S\) then \(\iint_{S} u \nabla u \cdot \mathbf{n} d S=\iiint_{D}|\nabla u|^{2} d V\)

Let \(S\) be the cylinder \(x^{2}+y^{2}=a^{2},\) for \(-L \leq z \leq L\) a. Find the outward flux of the field \(\mathbf{F}=\langle x, y, 0\rangle\) across \(S\) b. Find the outward flux of the field \(\mathbf{F}=\frac{\langle x, y, 0\rangle}{\left(x^{2}+y^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\) across \(S\), where \(|\mathbf{r}|\) is the distance from the \(z\) -axis and \(p\) is a real number. c. In part (b), for what values of \(p\) is the outward flux finite as \(a \rightarrow \infty(\text { with } L\) fixed)? d. In part (b), for what values of \(p\) is the outward flux finite as \(L \rightarrow \infty\) (with \(a\) fixed)?

Consider the rotational velocity field \(\mathbf{v}=\langle-2 y, 2 z, 0\rangle\) a. If a paddle wheel is placed in the \(x y\) -plane with its axis normal to this plane, what is its angular speed? b. If a paddle wheel is placed in the \(x z\) -plane with its axis normal to this plane, what is its angular speed? c. If a paddle wheel is placed in the \(y z\) -plane with its axis normal to this plane, what is its angular speed?
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