Problem 37
Question
Describe geometrically the level surfaces for the functions. \(f(x, y, z)=4 x^{2}-9 y^{2}\)
Step-by-Step Solution
Verified Answer
The level surfaces are cylindrical shapes with hyperbolic cross-sections in the \(xy\)-plane.
1Step 1: Understanding the Function
The given function is a function of three variables: \(f(x, y, z) = 4x^2 - 9y^2\). This function, however, is not dependent on \(z\), meaning its level surfaces will be constant for any \(z\) value. This indicates that the level surfaces will be cylindrical since the \(z\) dimension doesn't affect the surface.
2Step 2: Setting Up the Level Surface Equation
To find the level surfaces, we set \(f(x, y, z) = c\) for some constant \(c\). So the equation becomes: \[4x^2 - 9y^2 = c.\] This is the general form of a hyperboloid equation when considering \(x\) and \(y\).
3Step 3: Analyzing Different Cases for c
Depending on the value of \(c\), the shapes of the level surfaces change. - If \(c = 0\), the equation becomes \(4x^2 = 9y^2\), which represents two intersecting lines in the \(xy\)-plane.- If \(c > 0\), the equation \(4x^2 = 9y^2 + c\) describes a hyperbola centered at the origin in the \(xy\)-plane.- If \(c < 0\), rearranging gives \(9y^2 = 4x^2 - c\), which is again a hyperbola, but the orientation is different from when \(c > 0\).
4Step 4: Interpreting Geometric Shapes for c
Since these level surfaces do not contain \(z\), for each value of \(c\), the surface extends infinitely in the \(z\) direction, forming a cylindrical shape. Thus, geometrically, these are cylinders with hyperbolic cross-sections in the \(xy\)-plane.
Key Concepts
Geometric InterpretationMultivariable CalculusHyperboloid EquationCylindrical Surfaces
Geometric Interpretation
Understanding the geometric interpretation of level surfaces can provide a clearer insight into their shapes and structures. In this context, the level surfaces originate from setting the function of three variables, \( f(x, y, z) = 4x^2 - 9y^2 \), equal to a constant. These can be visualized as surfaces cutting through the three-dimensional space.
For this specific function, the level surfaces are independent of the variable \(z\). This means their structure remains unchanged regardless of \(z\)'s value. Imagine slicing a loaf of bread where each slice perfectly matches the previous one in form. Similarly, our surfaces are uniform along the \(z\) direction because the equation only involves \(x\) and \(y\).
In simpler terms, these level surfaces don't "care" about \(z\) and are thus infinitely extended along this axis. The result is essentially a series of two-dimensional shapes - namely, hyperbolas and lines - that stretch infinitely through the third dimension, forming what we call cylindrical surfaces. The geometry is pivotal because it conveys the boundless nature of these 2D patterns when extended into 3D space.
For this specific function, the level surfaces are independent of the variable \(z\). This means their structure remains unchanged regardless of \(z\)'s value. Imagine slicing a loaf of bread where each slice perfectly matches the previous one in form. Similarly, our surfaces are uniform along the \(z\) direction because the equation only involves \(x\) and \(y\).
In simpler terms, these level surfaces don't "care" about \(z\) and are thus infinitely extended along this axis. The result is essentially a series of two-dimensional shapes - namely, hyperbolas and lines - that stretch infinitely through the third dimension, forming what we call cylindrical surfaces. The geometry is pivotal because it conveys the boundless nature of these 2D patterns when extended into 3D space.
Multivariable Calculus
In multivariable calculus, we frequently deal with functions that depend on more than one variable. Here, our focus lies in visualizing such functions through their level surfaces. These are crucial as they furnish a transparent as well as a visual understanding of the behavior of functions having multiple inputs, such as \( f(x, y, z) = 4x^2 - 9y^2 \).
Level surfaces offer insights into how these variables interact. By setting \( f(x, y, z) = c \), where \(c\) is a constant, we're essentially exploring all the three-dimensional points \((x, y, z)\) that result in the same function value.
This approach simplifies analyzing such functions since it essentially reduces a 3D problem into manageable cross-sections, making it easier to interpret the function behavior. In calculus, this becomes especially important when dealing with optimization problems, surface integrals, or understanding the topology of functions across different dimensions.
Level surfaces offer insights into how these variables interact. By setting \( f(x, y, z) = c \), where \(c\) is a constant, we're essentially exploring all the three-dimensional points \((x, y, z)\) that result in the same function value.
This approach simplifies analyzing such functions since it essentially reduces a 3D problem into manageable cross-sections, making it easier to interpret the function behavior. In calculus, this becomes especially important when dealing with optimization problems, surface integrals, or understanding the topology of functions across different dimensions.
Hyperboloid Equation
The equation \(4x^2 - 9y^2 = c\) is derived from the function \(f(x, y, z)\) to describe level surfaces. It's a form of a hyperboloid equation when analyzed in the context of \(x\) and \(y\). This equation is essential in understanding how the level surfaces manifest in space.
Different values of \(c\) define various shapes:
Different values of \(c\) define various shapes:
- If \(c = 0\), the equation simplifies to \(4x^2 = 9y^2\), which geometrically corresponds to two intersecting straight lines lying in the \(xy\)-plane.
- For \(c > 0\), the equation represents a hyperbola, opening along the axes within the \(xy\)-plane, seen as hyperbolic curves.
- When \(c < 0\), the geometry is still hyperbolic, with a different orientation indicating another set of hyperbolas intersecting the plane.
Cylindrical Surfaces
A cylindrical surface in mathematics is a surface that extends parallel to a particular axis - in this case, the \(z\)-axis. For our function, \(4x^2 - 9y^2 = c\), the presence of cylindrical surfaces is due to the fact that the function does not depend on \(z\).
The level surface equation is unconditional regarding \(z\), resulting in a repetition of the same two-dimensional shape, a hyperbola or straight lines, along the entire \(z\) dimension.
Geometrically, for each \(c\), this means:
The level surface equation is unconditional regarding \(z\), resulting in a repetition of the same two-dimensional shape, a hyperbola or straight lines, along the entire \(z\) dimension.
Geometrically, for each \(c\), this means:
- The surface extends infinitely upwards and downwards along the \(z\)-axis.
- Each cross-section parallel to the \(xy\)-plane portrays the hyperbolic or linear pattern defined by \(4x^2 - 9y^2 = c\).
Other exercises in this chapter
Problem 37
Let \(f(x, y)=x^{2} y /\left(x^{4}+y^{2}\right)\). (a) Show that \(f(x, y) \rightarrow 0\) as \((x, y) \rightarrow(0,0)\) along any straight line \(y=m x\). (b)
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Express the following in \(\partial\) notation. (a) \(f_{y y y}\) (b) \(f_{x x y}\) (c) \(f_{x y y y}\)
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A steel box without a lid having volume 60 cubic feet is to be made from material that costs \(\$ 4\) per square foot for the bottom and \(\$ 1\) per square foo
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Express the following in subscript notation. (a) \(\frac{\partial^{3} f}{\partial x^{2} \partial y}\) (b) \(\frac{\partial^{4} f}{\partial x^{2} \partial y^{2}}
View solution