Question

Sketch the graph of the level surface \(f(x, y, z)=c\) at the given value of \(c\) $$ f(x, y, z)=\sin x-z, \quad c=0 $$

Short answer

The level surface \(f(x, y, z)=0\) for the given function \(f(x, y, z)=\sin x - z\) results in the equation \(z = \sin x\). This describes a wave pattern in 3D space, where \(z\) varies from -1 to 1 as a wave along the \(x\)-axis, and the pattern is extended along the \(y\)-axis as \(y\) does not affect our level surface.

Step-by-step solution

Understand and Setup the Equation

The equation given by the problem is \(f(x, y, z)= \sin x - z\). A level surface is the set of all points \( (x, y, z) \) for which \(f(x, y, z) = c\) is a constant. The equation for our level surface is \(\sin x - z = 0\) as \(c = 0\). This is our starting equation.

Rearrange the Equation and Find the Relationship

If we rearrange the equation of our level surface, we get \(z = \sin x\). This is the relationship between \(x\) and \(z\). The value of \(y\) can be any real number as \(y\) is not present in our equation and hence will not affect the level surface. Thus, each point on the level surface has the form \((x,y, \sin x)\).

Sketch the Graph

Our sketch should represent each point in 3D space that satisfies \(z = \sin x\). It's important to remember that the function \( \sin x \) varies between -1 and 1. Thus, our sketch will appear as a wave with peaks at 1 and valleys at -1. The wave will vary with \(x\), whereas \(y\) can take any value. Due to this, our sketch will have repeated waves of \( \sin x \) on the z-axis spreading along the y-axis.

Key concepts

Multivariable Calculus

Multivariable calculus extends the principles and methods of calculus to functions of more than one variable. In essence, while basic calculus focuses on rates of change and accumulation in functions along a single dimension, multivariable calculus concerns itself with space that is two-dimensional and beyond.

Specifically, this field of study examines how we can analyze curves, surfaces, and volumes in three-dimensional space and handles concepts such as partial derivatives, multiple integrals, and, relevant to our exercise, level surfaces.

A level surface is a three-dimensional analog to a level curve (also known as a contour line), representing a set of points where a function has the same value—that constant value being denoted typically by 'c'. When the domain of the function is three-dimensional—covering variables x, y, and z—these level surfaces become a powerful way to visualize relationships between the variables in a spatial context.

3D Graph Sketching

3D graph sketching is a technique used to visualize functions of two variables as surfaces in three-dimensional space. To accurately sketch a 3D graph, it's crucial to understand the interplay between the variables involved.

A typical 3D graph has an x-axis, y-axis, and z-axis, with the given function determining the shape of the surface in this coordinate system. When sketching level surfaces, you're essentially drawing a 'slice' of this three-dimensional graph corresponding to a specific function value.

It requires understanding shapes and behaviors common to the functions involved, such as recognising the appearance of sine waves or cylinders in the context of our given exercise. Even though this is not always easy, having a clear step-by-step approach can greatly aid in constructing accurate and insightful representations of the math at play.

Sine Function

The sine function, denoted as sin, is a fundamental trigonometric function that describes a smooth periodic oscillation, like the movement of a wave. For a real number x, which often represents an angle, the sine function provides the ratio of the length of the side of a right-angled triangle opposite the angle to the length of the triangle's hypotenuse.

When graphed, the sine function produces a characteristic 'wave-like' shape, smoothly rising to a maximum of 1 and descending to a minimum of -1, which are its peaks and valleys, hence making it periodic with a period of \(2\pi\).

In the context of multivariable calculus and 3D graph sketching, the sine function is essential for depicting natural phenomena displaying wave patterns or oscillatory behavior, such as ripples on a water surface or sound waves, within a multivariable framework.

Constant Function

A constant function is a function that always returns the same value no matter what the input is. In other words, it's a horizontal line on a two-dimensional graph. With respect to multivariable calculus, when we set a function equal to a constant, we are interested in finding the set of points that satisfy this condition, creating what is referred to as a level surface.

In three dimensions, if a function could be thought of as varying in height across a landscape of coordinates, a constant function would represent a flat horizontal plane at that height.

Bringing this back to our exercise, when the function \(f(x, y, z) = \sin x - z\) is set equal to 0, a constant, we are looking for all points in space that lie on a surface where the value of the function is fixed at 0. This constraint helps us to visualize and sketch the level surface effectively.