/*! 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 21 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 21

Sketch the following vector fields. $$\mathbf{F}=\langle 1,0, z\rangle$$

Short Answer

Expert verified
Question: Based on the given vector field, \(\mathbf{F} = \langle 1, 0, z \rangle\), describe the behavior of the vectors in space. Answer: The vectors in the given vector field always point in the positive x-direction with a magnitude of 1 considering only the x-component since \(F_x = 1\). They are confined to the x-z plane as there is no change in the y-direction due to \(F_y = 0\). The z-component of the vectors depends on the z-coordinate of the points in space, as \(F_z = z\).

Step by step solution

01

Identify the components of the vector field

Begin by identifying the components of the vector field, \(\mathbf{F}=\langle 1,0,z \rangle\). In this case, the vector field has the following components: $$F_x = 1, \ F_y = 0, \ F_z = z.$$
02

Analyze the behavior of the vector field

From the components of the vector field, one can analyze its behavior in space based on \(F_x\), \(F_y\), and \(F_z\): - \(F_x = 1\): No matter the position in space, \(x\) component is always 1, so the vectors will always point in the positive x-direction with a magnitude of 1 considering only the x-component. - \(F_y = 0\): This means that there is no change in the \(y\)-direction in any location in the space. Therefore, the vectors are confined to the x-z plane and do not extend into the y-direction. - \(F_z = z\): The magnitude of \(z\) component depends on the \(z\)-coordinate of the point in space. It simply means that the vector follows the z-value of the point which makes its z-component equal to the \(z\)-coordinate of the point in space.
03

Sketch the vector field

Since these vectors are confined to the x-z plane and have a fixed \(x\) component equal to 1, one can begin by sketching random points with z-values in the x-z plane. For each point, draw a vector that points in the positive x-direction, with a magnitude of 1 because of \(F_x = 1\). Position this tail at the given point, and ensure that the z-component of the vector matches the z-coordinate of the point. Repeat this process for several points in the x-z plane to create a representative sketch of the vector field \(\mathbf{F} = \langle 1, 0, z \rangle\).

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Key Concepts

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

Vector Field Components
Understanding the components of a vector field is a fundamental step in sketching and analyzing vector fields. Each vector within the field is defined by components that determine how it behaves at any given point in space. With regard to the vector field \( \mathbf{F} = \langle 1, 0, z \rangle \), the three components are:\
\
\
    \
  • \(F_x = 1\): represents a constant effect in the positive x-direction,\
    • \
  • \(F_y = 0\): indicates no movement or force in the y-direction,\
    • \
  • \(F_z = z\): implies that the impact in the z-direction varies linearly with the z-coordinate itself.\
    • \
      \
      \
      \By grasping these components, one can infer that the vectors across our field will maintain a reliable and unvarying influence towards the positive x-axis, while the influence in the z-direction will differ depending on the location within the field. Understanding this consistent behavior in the x-direction while simultaneously appreciating the variable nature of the z-component is key to effectively sketching the field.
Vector Field Behavior
The behavior of a vector field represents how the vectors function and interact within the domain of the field. For instance, in the vector field \( \mathbf{F} = \langle 1, 0, z \rangle \) analyzed in the exercise, each vector's action is unique to its position on the x-z plane.

Digging deeper into the behavior of our field, you'll note:\
    \
  • The constant x-component (\(F_x = 1\)) suggests a uniform force or flow in the positive x-direction across the entire field.\
    • \
  • The z-component (\(F_z = z\)) escalates or de-escalates in magnitude with the z-coordinate. This variability contributes to a stronger upward or downward directionality as we move away from the x-axis (z=0).\
    • \
  • Given the zero y-component (\(F_y = 0\)), the vectors never veer off into the y-axis, maintaining the field's confinement to the x-z plane.\
    • \
      \
      \
      \In essence, to fully grasp the vector field's behavior, visualizing how these vectors change鈥攐r do not change鈥攁cross various points is key. Understanding that in our field, every point has a consistent push in the x-direction and a z-directed force proportional to its own z-value, renders sketching and predicting the field's physical manifestations considerably more intuitive.
3D Vector Plotting
Plotting a vector field in three dimensions allows us to visualize the orientation and magnitude of the vectors at various points in space. 3D plotting is particularly powerful for fields like \( \mathbf{F} = \langle 1, 0, z \rangle \) that have non-uniform components, offering insights into the field鈥檚 complex structure that might not be apparent in two-dimensional representations.

When plotting our vector field, start by:
  • Selecting a range of values for each coordinate axis to define the scope of the plot.
  • Calculating the components of the vector at each point within this range.
  • Sketching the vectors according to their components, paying close attention to maintaining the correct direction and length proportions as represented by the respective components, \(F_x\), \(F_y\), and \(F_z\), for each vector.


      • For a successful sketch, consider using a grid or 3D graphing software to accurately represent the vectors. In the field \( \mathbf{F} = \langle 1, 0, z \rangle \), vectors are always pointing somewhat to the positive x-axis but with varying degrees of intensity in the z-direction, dependent on their z-coordinate. Its simplicity provides a solid foundation for students beginning to understand the complex visuals that 3D vector plotting can exhibit.

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Most popular questions from this chapter

The Navier-Stokes equation is the fundamental equation of fluid dynamics that models the flow in everything from bathtubs to oceans. In one of its many forms (incompressible, viscous flow), the equation is $$\rho\left(\frac{\partial \mathbf{V}}{\partial t}+(\mathbf{V} \cdot \nabla) \mathbf{v}\right)=-\nabla p+\mu(\nabla \cdot \nabla) \mathbf{V}$$ In this notation, \(\mathbf{V}=\langle u, v, w\rangle\) is the three-dimensional velocity field, \(p\) is the (scalar) pressure, \(\rho\) is the constant density of the fluid, and \(\mu\) is the constant viscosity. Write out the three component equations of this vector equation. (See Exercise 40 for an interpretation of the operations.)

Zero circulation fields. Consider the vector field \(\mathbf{F}=\langle a x+b y, c x+d y\rangle .\) Show that \(\mathbf{F}\) has zero circulation on any oriented circle centered at the origin, for any \(a, b, c,\) and \(d,\) provided \(b=c\)

One of Maxwell's equations for electromagnetic waves is \(\nabla \times \mathbf{B}=C \frac{\partial \mathbf{E}}{\partial t},\) where \(\mathbf{E}\) is the electric field, \(\mathbf{B}\) is the magnetic field, and \(C\) is a constant. a. Show that the fields \(\mathbf{E}(z, t)=A \sin (k z-\omega t) \mathbf{i}\) and \(\mathbf{B}(z, t)=A \sin (k z-\omega t) \mathbf{j}\) satisfy the equation for constants \(A, k,\) and \(\omega,\) provided \(\omega=k / C\). b. Make a rough sketch showing the directions of \(\mathbf{E}\) and \(\mathbf{B}\).

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.

Alternative construction of potential functions Use the procedure in Exercise 71 to construct potential functions for the following fields. $$\quad \mathbf{F}=\langle-y,-x\rangle$$
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