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Chapter 3: Problem 2

Skizzieren Sie das ebene Vektorfeld \(\vec{v}(x, y)=(1, \sin x)\).

Short Answer

Expert verified
The vector field \( \vec{v}(x, y) = (1, \sin x) \) shows horizontal vectors with y-components oscillating in a wave-like pattern due to the sine of x.

Step by step solution

01

Understanding the Vector Field

The given vector field \( \vec{v}(x, y) = (1, \sin x) \) means that for each point \((x, y)\) in the plane, there is a vector \( (1, \sin x) \) associated with it. This vector field has a constant component of 1 in the x-direction and a sine-modulated component in the y-direction.
02

Analyzing x-Component

The x-component of the vector, which is 1, indicates that every vector in the field has the same length of 1 in the x-direction. This represents a uniform flow in the positive x-direction across the entire plane.
03

Analyzing y-Component

The y-component is \( \sin x \), meaning the length of the vector in the y-direction depends on the sine of the x-coordinate. The sine function oscillates between -1 and 1, influencing the direction and magnitude of the vector along the y-axis.
04

Sketching the Vector Field

To sketch \( \vec{v}(x, y) = (1, \sin x) \), draw vectors at various points \((x, y)\). Each vector should point 1 unit in the positive x-direction and \( \sin x \) units in the y-direction. Note that when \( x = 0, \pi, 2\pi, ... \) (multiples of \( \pi \)), the y-component is 0; at these points, vectors are horizontal. Vectors oscillate upwards when \( x = \frac{\pi}{2}, \frac{5\pi}{2}, ... \) and downwards when \( x = \frac{3\pi}{2}, \frac{7\pi}{2}, ... \).
05

Observing Patterns

Observe that the vector field has a wave-like pattern in the y-direction due to the sine function. As x increases, the vectors oscillate between pointing slightly upwards and slightly downwards while always moving in the positive x-direction.
06

Completing the Sketch

Ensure the sketch covers a few periods of \( \sin x \) to capture the repetitive nature of the field. All vectors should have the same horizontal component, demonstrating that they follow a consistent left-to-right flow.

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Key Concepts

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

Oscillation Patterns
In vector field analysis, understanding oscillation patterns is crucial. Oscillation describes how a function, like sine, moves up and down in a periodic manner. This can be described as a repetitive movement. For the vector field \( \vec{v}(x, y) = (1, \sin x) \), this pattern is evident in the y-component of the vectors.
The sine function creates a wave-like motion in the y-direction. This means as you move along the x-axis:
  • The y-component of each vector cycles up and down.
  • Points such as \( x = \pi, 2\pi, 3\pi, \ldots \) are when the y-component is zero and align flat along the x-axis.
  • At these points, the vectors are horizontal.
This oscillatory behavior creates an interesting and regular pattern in our vector field, an excellent visual representation of periodic functions.
Sine Function Behavior
The sine function is pivotal in understanding the behavior of the vector field \( \vec{v}(x, y) = (1, \sin x) \). The sine function, \( \sin x \), oscillates between -1 and 1. This behavior directly affects the y-component of the vectors in the field.
Here's how this affects the vector field:
  • At certain points, such as \( x = 0, \pi, 2\pi \), the function value is 0. Vectors align horizontally here because the y-component is zero.
  • When \( x = \frac{\pi}{2}, \frac{5\pi}{2}, \ldots \), \( \sin x = 1 \), causing vectors to point upwards.
  • Conversely, when \( x = \frac{3\pi}{2}, \frac{7\pi}{2}, \ldots \), \( \sin x = -1 \), meaning vectors point downwards.
Understanding the periodic and predictable nature of the sine function helps with sketching and analyzing the vector field's oscillation pattern.
Vector Components
Vectors consist of components that determine their direction and magnitude. For \( \vec{v}(x, y) = (1, \sin x) \), each vector has:
  • A constant x-component of 1, indicating uniform movement in the positive x-direction.
  • A variable y-component determined by \( \sin x \), causing the y-direction to fluctuate according to the sine wave behavior.

These components combine to form vectors that have:
  • A steady horizontal movement due to the constant x-component.
  • Vertical oscillation as influenced by the sine-modulated y-component.
By breaking down vectors into their components, it's easier to visualize how they behave in the field.
Sketching Vector Fields
Sketching a vector field is a way to visualize how vectors behave over an area. For the field \( \vec{v}(x, y) = (1, \sin x) \), it's about plotting vectors at various points, considering both their direction and magnitude.
To begin sketching:
  • Plot vectors with a consistent horizontal component (1 unit to the right), reflecting the uniform x-component.
  • Adjust the vertical component according to \( \sin x \), oscillating between -1 and 1.
When sketching:
  • Ensure the vectors along \( x = 0, \pi, 2\pi, \ldots \) remain horizontal.
  • Capture the upwards movement at peaks (where \( \sin x = 1 \)) and downwards at troughs (where \( \sin x = -1 \)).
This approach results in a clear visual pattern, showcasing both the constant flow in the x-direction and the oscillating behavior in the y-direction.

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Most popular questions from this chapter

Berechnen Sie \(^{c} \int \vec{v} \mathrm{~d} \vec{r}\) f眉r \(\vec{v}=\left(2 x-y,-y^{2} z^{2}, x y z\right)\) und \(C \cdot \vec{r}(t)=(\cos t, \sin t, 0), 0 \leqq t \leqq 2 \pi\). Was f眉r eine \(\mathrm{K}\) urve ist \(C\) ? Ist \(\vec{v}\) konservativ?

Es sei \(\vec{v}=(2 y+3, x z, y z-x)\). Man berechne \(^{c} \int \vec{v} \mathrm{~d} \vec{r}\) f眉r a) \(C: \vec{r}(t)=\left(2 t^{2}, t, t^{3}\right), 0 \leqq t \leqq 1\) b) \(C\) : die Strecke mit demselben Anfangs- und Endpunkt wie die Kurve aus a).

Berechnen Sie den Schwerpunkt einer homogenen Halbkugel.

Ein geschlossener Zylinder mit kreisformigem Querschnitt ist mit einem Pulver gef眉llt. Das Gef盲B rotiert um scine Achse. Dabei verdichtet sich das Pulver an der Mantelfl盲che, und zwar sei die Dichte im Abstand \(r\) von der Achse \(c^{\circ}\left(r^{2}+r_{0}^{2}\right)\). Welches Tr?gheitsmoment hat das Pulver? \(\left(c\right.\) und \(r_{0}\) seien positive Zahlen).

Berechnen Sie das Tr盲gheitsmoment eines Kreiszylinders konstanter Massendichte \(\rho\) bez. einer auf seiner Mantelff盲che liegenden Drehachse.
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