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Chapter 10: Problem 8

Describe and sketch the surface. \(z=\sin y\)

Short Answer

Expert verified
The surface is an infinite sinusoidal wave extending parallel to the x-axis.

Step by step solution

01

Understand the Function

The given equation is \( z = \sin y \). Here, \( z \) depends only on \( y \), and there is no \( x \) variable in the equation. This implies that any vertical plane parallel to the \( yz \)-plane (where \( x \) is constant) will reproduce the same sinusoidal curve.
02

Sketch the Sinusoidal Curve in 2D

First, let's consider a 2D plane where \( x = 0 \), which reduces the equation to \( z = \sin y \). Sketch the sinusoidal curve for this equation. The sinusoid oscillates between -1 and 1 with a periodicity of \( 2\pi \).
03

Extend to 3D Surface

Since \( z = \sin y \) is independent of \( x \), this curve is the same for any value of \( x \). Thus, extend the sinusoidal curve \( z = \sin y \) infinitely in the \( x \)-direction, creating a wave-like surface parallel to the \( yz \)-plane.
04

Visualize the Surface

Visualize this as a series of parallel waves extending infinitely in the \( x \) direction. Imagine ripples in water that move along the \( y \)-axis direction uniformly without varying with \( x \). Each slice of the surface parallel to the \( yz \)-plane is identical.

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Key Concepts

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

Sinusoidal Curves
In the study of sinusoidal curves, the function \( z = \sin y \) creates a cyclical pattern in mathematics that represents repetitive oscillations. In this case, \( z \) varies distinctly with \( y \), forming peaks and troughs repeating regularly. Such curves demonstrate the inherent properties of sine functions:
  • Amplitude: The maximum and minimum values the curve reaches, here \( z \) oscillates between -1 and 1.
  • Periodicity: Refers to the length required for the curve to repeat itself, which is \( 2\pi \) for sine functions.
  • Waveform Shape: It exhibits smooth, continuous, wave-like patterns.
Understanding these basic features aids in predicting how the curve behaves over any interval. Sinusoids are foundational in various fields, including engineering and physics, reflecting fundamental wave phenomena.
3D Visualization
Visualizing mathematical functions in three dimensions can enhance your comprehension of their behavior. For the equation \( z = \sin y \), think of extending the sinusoidal curve along the \( x \)-axis.

Here鈥檚 how to imagine it:
  • Imagine a 2D sinusoidal wave and stretch it across the third axis, forming a surface with repeating wave patterns.
  • Each slice parallel to the \( yz \)-plane appears as a familiar sine wave.
  • This surface extends infinitely in the \( x \)-direction, like a fabric woven by continuous waves.
This method of visualization not only enhances your spatial reasoning but gives you better insight into how mathematical surfaces behave in space. Such visualization techniques are crucial for fields like computer graphics and physics where understanding surface interactions is vital.
Mathematical Surfaces
Mathematical surfaces describe complex shapes formed by equations like \( z = \sin y \). These surfaces can help model real-world phenomena such as waves, terrain, or sound.

Key characteristics to note include:
  • Uniformity: The "surface" resembles parallel waves, consistent across the entire 3D space since \( z \) remains unaffected by \( x \).
  • Dimensionality: The surface is defined in a 3D space, showcasing how equations irrespective of one variable can manifest in different forms.
  • Application: Such surfaces provide a foundation for modeling and simulation in environmental science, engineering, and art.
By comprehending mathematical surfaces, you gain analytical skills essential for visualizing and interpreting data or creating accurate models in various technological and scientific applications.

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