/*! 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 19 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 19

Sketch the following vector fields. $$\mathbf{F}=\left\langle e^{-x}, 0\right\rangle$$

Short Answer

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Question: Sketch the vector field: $$\mathbf{F}=\left\langle e^{-x}, 0\right\rangle$$ Answer: The vector field is horizontally aligned with decreasing length as \(x\) increases and parallel to the x-axis. The \(y\)-component is always zero. See the provided sketch for reference.

Step by step solution

01

Constructing a table of values

For different values of \(x\) and \(y\), let's find the value of \(\mathbf{F}\). To simplify the process, we take a few points from the domain and compute the corresponding vectors. Here is the table of values for the vector field: | x | y | \(\mathbf{F} = \left\langle e^{-x}, 0 \right\rangle\) | |:----:|:-----:|:--------------------------------------------------:| | 0 | 0 | \(\left\langle1, 0\right\rangle\) | | 1 | 0 | \(\left\langle\frac{1}{e}, 0\right\rangle\) | | 2 | 0 | \(\left\langle\frac{1}{e^2}, 0 \right\rangle\) | | 0 | 1 | \(\left\langle1, 0 \right\rangle\) | | 1 | 1 | \(\left\langle\frac{1}{e}, 0 \right\rangle\) | | 2 | 1 | \(\left\langle\frac{1}{e^2}, 0 \right\rangle\) |
02

Drawing the vector field

Using the table from Step 1, we can now sketch the given vector field. 1. Start by drawing the Cartesian coordinate plane with x-axis and y-axis. 2. For each point in the table, draw an arrow from the selected point to the corresponding vector. 3. Label the arrows with the corresponding vector components. Make sure that the magnitude and direction of the arrows follow the values in the table. Notice that the vectors are aligned horizontally and decreasing in length as \(x\) increases. Also, note that the \(y\)-component is always zero, so the vectors stay parallel to the x-axis. The resulting vector field should look like this: ``` |----------------------------------> |--------------------------------> |------------------------------> |-------------> |--------> ---------------------------------------------------------- |----------------------------------> |_ _ _ _ _ _ _ _ _|_ _ _ _ _|_ _ _| |------------------------------> |------------|-------------|-------------|-------------| (x-axis) ```

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Key Concepts

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

Vector Calculus
Vector calculus is a branch of mathematics that focuses on vector fields and their applications. In physics and engineering, we often use vector calculus to describe and analyze physical phenomena such as electromagnetic fields and fluid flow.

In vector calculus, a **vector field** is a function that assigns a vector to each point in space. For example, the vector field \( \mathbf{F} = \left\langle e^{-x}, 0 \right\rangle \) assigns vectors based on the value of \( x \). The vector field describes how these vectors can change across different points on a plane.

  • The length or magnitude of the vectors can indicate intensity or strength at that point.
  • The direction of these vectors can illustrate the flow or direction of a field, such as how wind might blow across a region.
These properties are essential when sketching a vector field, as they dictate how each vector behaves and interacts with its surroundings.
Coordinate Plane
The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two perpendicular lines: the x-axis and y-axis. This system allows us to plot points, lines, and vectors precisely.

When studying vector fields, the coordinate plane provides a framework to visualize the behavior of vectors. In our example, \( \mathbf{F} = \left\langle e^{-x}, 0 \right\rangle \), all vectors lie on the coordinate plane and stay parallel to the x-axis. This specific vector field illustrates how vectors shrink in length as the x-value increases, which you can easily observe when plotted.

  • Origin: The starting point (0,0) is often used as a central reference.
  • X-axis: Horizontal line that helps indicate forward or backward direction.
  • Y-axis: Vertical line typically used for up or down direction.
Using the coordinate plane, we can consistently and accurately depict where each vector starts and how it moves.
Vector Components
Vector components break down a vector into its parts, illustrating direction and magnitude along the coordinate axes. For a 2D vector \( \mathbf{F} = \langle a, b \rangle \), \( a \) and \( b \) are the vector's components.

In the vector field \( \mathbf{F} = \left\langle e^{-x}, 0 \right\rangle \), the components are particularly simple:

  • X-Component: \( e^{-x} \) indicates how far the vector extends along the x-axis, decreasing as x increases.
  • Y-Component: 0 means there is no vertical movement; vectors do not rise or fall, maintaining a horizontal direction.
Understanding vector components is crucial when sketching or analyzing vector fields, as they define the shape and direction of each vector explicitly. In practical applications, identifying these components helps predict and understand the field鈥檚 behavior at any given point.

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Most popular questions from this chapter

Given a vector field \(\mathbf{F}=\langle f, 0\rangle\) and curve \(C\) with parameterization \(\mathbf{r}(t)=\langle x(t), y(t)\rangle,\) for \(a \leq t \leq b,\) we see that the line integral \(\int_{C} f d x+g d y\) simplifies to \(\int_{C} f d x\) a. Show that \(\int_{C} f d x=\int_{a}^{b} f(t) x^{\prime}(t) d t\) b. Use the vector field \(\mathbf{F}=\langle 0, g\rangle\) to show that \(\int_{C} g d y=\int_{a}^{b} g(t) y^{\prime}(t) d t\) c. Evaluate \(\int_{C} x y d x,\) where \(C\) is the line segment from (0,0) to (5,12) d. Evaluate \(\int_{C} x y d y,\) where \(C\) is a segment of the parabola \(x=y^{2}\) from (1,-1) to (1,1)

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. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

Integration by parts (Gauss' Formula) Recall the Product Rule of Theorem \(17.13: \nabla \cdot(u \mathbf{F})=\nabla u \cdot \mathbf{F}+u(\nabla \cdot \mathbf{F})\) a. Integrate both sides of this identity over a solid region \(D\) with a closed boundary \(S\), and use the Divergence Theorem to prove an integration by parts rule: $$\iiint_{D} u(\nabla \cdot \mathbf{F}) d V=\iint_{S} u \mathbf{F} \cdot \mathbf{n} d S-\iiint_{D} \nabla u \cdot \mathbf{F} d V$$ b. Explain the correspondence between this rule and the integration by parts rule for single-variable functions. c. Use integration by parts to evaluate \(\iiint_{D}\left(x^{2} y+y^{2} z+z^{2} x\right) d V\) where \(D\) is the cube in the first octant cut by the planes \(x=1\) \(y=1,\) and \(z=1\)

Zero circulation fields. For what values of \(b\) and \(c\) does the vector field \(\mathbf{F}=\langle b y, c x\rangle\) have zero circulation on the unit circle centered at the origin and oriented counterclockwise?

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$
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