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Chapter 15: Problem 7

Sketch the vector field by drawing some representative non intersecting vectors. The vectors need not be drawn to scale, but they should be in reasonably correct proportion relative to each other. $$ \begin{array}{l}{\mathbf{F}(x, y)=y \mathbf{i}-x \mathbf{j} . \text { [Note: Each vector in the field is per- }} \\ { \text { pendicular to the position vector }\mathbf{r}=x \mathbf{i}+y \mathbf{j} \cdot]}\end{array} $$

Short Answer

Expert verified
The vector field \( \mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j} \) represents circular field lines around the origin.

Step by step solution

01

Identify the Vector Field Components

The given vector field \( \mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j} \) consists of two components. The \( i \)-component is \( y \), which represents the horizontal part of the vector. The \( j \)-component is \( -x \), which represents the vertical part of the vector. This means that at any point \( (x, y) \), the vector points in the direction \( (y, -x) \).
02

Relationship with Position Vector

Notice that the vector \( \mathbf{F} \) is perpendicular to the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} \). This suggests that for any \( (x, y) \), the vector field lines should be tangent to a circle centered at the origin, since the vectors are perpendicular to the radius.
03

Sketching Guidelines

Begin by selecting a few points \( (x, y) \) on the plane. At each point, draw a vector in the direction \( (y, -x) \). The vectors should generally form a pattern that revolves around the origin, creating circular paths.
04

Drawing Vectors at Specific Points

For instance, at the point \( (1, 0) \), the vector will be \( (0, -1) \), pointing downward. At \( (0, 1) \), the vector will be \( (1, 0) \), pointing rightward. At \( (-1, 0) \), the vector will be \( (0, 1) \), pointing upward. At \( (0, -1) \), the vector will be \( (-1, 0) \), pointing leftward. Draw these vectors to form parts of a circle around the origin.

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Key Concepts

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

Vector Components
Understanding vector components is essential in analyzing a vector field like \( \mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j} \). In any vector expression, components are the individual elements that contribute to the vector's overall direction and magnitude.
  • The \( i \)-component is associated with the horizontal direction, whereas the \( j \)-component relates to the vertical motion of the vector.
  • For the vector field \( \mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j} \), the \( i \)-component is \( y \), which dictates the horizontal direction.
  • The \( j \)-component is \( -x \), influencing the vertical direction.
  • Thus, at any point \( (x, y) \) in this vector field, the vector points in the direction \( (y, -x) \).
Each component supports the overall vector by determining its unique direction based on the position \( (x, y) \). This allows students to understand how individual components influence the field's vector pattern.
Position Vector
The position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} \) gives a straightforward way of locating any point \( (x, y) \) on the plane.
  • This vector originates from the coordinate system's origin and points directly to the coordinates \( (x, y) \).
  • The position vector is crucial for understanding how other vectors relate to the point \( (x, y) \).
  • Position vectors help to identify the relative orientation of vectors in a field, such as the one given in the exercise.
It serves as a reference point, aligning vectors in calculations and identifying whether they are tangent, parallel, or perpendicular to other vectors or lines.
Perpendicular Vectors
Perpendicular vectors have a unique relationship that affects vector field sketching. Vectors are perpendicular when the dot product between them equals zero.

Understanding Perpendicular Vectors

  • For the given vector field \( \mathbf{F}(x, y) = y \mathbf{i} - x \mathbf{j} \), it is noted that the vectors are perpendicular to the position vector \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} \).
  • The vectors in this field are tangent to circular paths centered at the origin, since they form right angles with the radii of these circles.
  • Mathematically, two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \) are perpendicular if \( a_1 b_1 + a_2 b_2 = 0 \).
Since the vector \( \mathbf{F} \) and the position vector \( \mathbf{r} \) are perpendicular, this is why the vector field \( \mathbf{F} \) forms circular patterns around the origin.

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Most popular questions from this chapter

Use Green鈥檚 Theorem to find the work done by the force field F on a particle that moves along the stated path. \(\mathbf{F}(x, y)=\sqrt{y} \mathbf{i}+\sqrt{x} \mathbf{j} ;\) the particle moves counterclockwise one time around the closed curve given by the equations \(y=0, x=2,\) and \(y=x^{3} / 4\)

Suppose that \(\mathbf{F}(x, y)=f(x, y) \mathbf{i}+g(x, y) \mathbf{j}\) is a vector field whose component functions \(f\) and \(g\) have continuous first partial derivatives. Let \(C\) denote a simple, closed, piecewise smooth curve oriented counterclockwise that bounds a region \(R\) contained in the domain of \(\mathbf{F}\). We can think of \(\mathbf{F}\) as a vector field in 3 -space by writing it as $$ \mathbf{F}(x, y, z)=f(x, y) \mathbf{i}+g(x, y) \mathbf{j}+0 \mathbf{k} $$ With this convention, explain why $$ \int_{C} \mathbf{F} \cdot d \mathbf{r}=\iint_{R} \operatorname{curl} \mathbf{F} \cdot \mathbf{k} d A $$

Evaluate the surface integral $$ \iint_{\sigma} f(x, y, z) d S $$ \(f(x, y, z)=\left(x^{2}+y^{2}\right) z ; \sigma\) is the portion of the sphere \(x^{2}+y^{2}+z^{2}=4\) above the plane \(z=1\)

Find the flux of the vector field \(\mathbf{F}\) across \(\sigma\) \(\mathbf{F}(x, y, z)=y \mathbf{j}+\mathbf{k} ; \sigma\) is the portion of the paraboloid \(z=x^{2}+y^{2}\) below the plane \(z=4,\) oriented by downward unit normals.

Confirm that the force field \(\mathbf{F}\) is conservative in some open connected region containing the points \(P\) and \(Q,\) and then find the work done by the force field on a particle moving along an arbitrary smooth curve in the region from \(P\) to \(Q .\) $$ \mathbf{F}(x, y)=e^{-y} \cos x \mathbf{i}-e^{-y} \sin x \mathbf{j} ; \quad P(\pi / 2,1), Q(-\pi / 2,0) $$
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