Question

In Exercises \(1-8,\) describe and sketch the surface. $$ z-\sin y=0 $$

Short answer

The given equation represents an infinite wave-like surface, created by repeating a sine curve in the x direction. The surface oscillates between -1 and 1 as it follows the pattern of a sine wave in the yz-plane, repeated for all values of x.

Step-by-step solution

Simplify Surface Equation

Simplify the equation from \(z-\sin y=0\) to \(z=\sin y\). This form helps us to better realize that this is the equation of a sine wave.

Sketch a 2D Sine Wave

Draw a regular 2D sine wave from the equation \(z=\sin y\). For the sketch, take y-axis as the horizontal axis and z-axis as the vertical axis. The wave oscillates between -1 and 1 because that's the range of the sine function. Hence, peaks will be at \(z=1\) and valleys will be at \(z=-1\).

Extend Sine Wave to 3D

Now extend this 2D sine wave into the third dimension by considering the \(x\) direction. Since \(x\) does not appear in the equation, \(x\) has no effect on the value of \(z\). The 2D sine wave we drew earlier is just repeated infinitely in the \(x\) direction, resulting in a wave surface extending infinitely in the \(x\) direction.

Key concepts

Sine Wave

A sine wave is a mathematical curve that describes a smooth, periodic oscillation. It is the graph of the sine function when plotted against an angle that progresses over time or space. The standard equation for a sine wave in two dimensions is given by \( z = \text{sin}(y) \), where \( y \) is typically considered as the horizontal axis and \( z \) as the vertical axis.

Understanding sine waves is crucial for grasping various concepts in mathematics and physics, such as sound waves, light waves, and even quantum mechanics. It's also foundational for understanding more complex waves and oscillations in 3D calculus. In our case, the sine wave is used to describe a 3D surface, where the values of \( z \) oscillate sinusoidally as \( y \) varies, creating peaks at \( z = 1 \) and valleys at \( z = -1 \). The key characteristics of a sine wave include its amplitude, the maximum value of oscillation (which is 1 in the standard sine function), its wavelength, the distance over which the wave's shape repeats (here, \( 2\text{π} \)), and its frequency, the number of wavelengths per unit distance.

3D Surfaces in Calculus

In calculus, especially multivariable calculus, the concept of 3D surfaces is fundamental for visualizing and understanding how two-dimensional functions extend into three dimensions. A 3D surface can be represented by an equation involving three variables, usually \( x, y, \) and \( z \).

The surface described by the equation \( z = \text{sin}(y) \) is a classic example where the third dimension \( x \) is independent of the other two, indicating a translation symmetry along the \( x \)-axis. This implies that for any slice of the surface taken parallel to the \( yz \)-plane, the cross-section will be a perfect sine wave.

3D surfaces can represent real-world phenomena like geographical landscapes, fluid dynamics, electromagnetic fields, and more. To study these surfaces, one may need to understand concepts like level sets, gradients, and directional derivatives. Such knowledge enables deeper insight into the behavior of functions and their applications to real-life problems.

Sketching Curves

Sketching curves is an essential skill in the mathematics field, often used to convey complex ideas through a visual medium. For functions of one or two variables, sketches are usually limited to 2D. However, when we look into functions that involve three variables, sketching can involve creating 3D representations.

When sketching the curve for our sine wave surface, \( z = \text{sin}(y) \), we start with a standard 2D sine wave graph. Then, to extend this into three dimensions, we translate the 2D curve along the \( x \)-axis. Since the equation does not depend on \( x \), the sine wave remains constant for all values of \( x \), creating a 'sine wave' effect that ripples along the \( yz \)-plane, extending indefinitely in both directions along the \( x \)-axis.

Effective sketching of curves can aid in better understanding the nature of the function and predicting the behavior of complex systems. It is often one of the first steps a student learns in multivariable calculus before delving into more abstract and analytical methods of studying these surfaces.