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Chapter 15: Problem 85

What symmetry will you find in a surface that has an equation of the form \(r=f(z)\) in cylindrical coordinates? Give reasons for your answer.

Short Answer

Expert verified
The surface is symmetrical about the z-axis due to independence from \( \theta \).

Step by step solution

01

Understanding the Equation

In cylindrical coordinates, the general form of a surface is given by the equation \( r = f(z) \), where \( r \) is the radial distance and \( z \) is the vertical axis. This equation does not include the angular coordinate \( \theta \).
02

Analyzing Symmetry Around Axis

Since the equation \( r = f(z) \) does not depend on \( \theta \), the surface is symmetrical about the \( z \)-axis. This means that for any fixed \( z \), \( r \) will be constant for all \( \theta \) values, creating a 'cylinder-like' symmetry.
03

Examining Additional Properties

Due to the independence from \( \theta \), rotating the surface around the \( z \)-axis will yield the same surface. This rotational symmetry confirms the axisymmetry of the surface.

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Key Concepts

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

Symmetry in Surfaces
Symmetry in surfaces refers to the property where a surface looks the same from different perspectives or after certain transformations. In cylindrical coordinates, equations can often exhibit various forms of symmetry, which simplifies analysis and understanding.

When an equation is of the form \(r = f(z)\), it lacks any dependency on the angular coordinate \(\theta\). This means that the surface generated by this equation remains unchanged if viewed from any angle around the \(z\)-axis. Essentially, the lack of \(\theta\) introduces a form of uniformity or sameness around that axis. Therefore, the surface exhibits symmetry about the \(z\)-axis. For each fixed \(z\), the radial distance \(r\) remains constant for all angles \(\theta\), leading to a perfectly balanced surface around the axis.
Axisymmetry
Axisymmetry is a specific type of symmetry. It involves a surface being symmetrical around a specific axis. In our example \(r = f(z)\), the symmetry is around the \(z\)-axis. This kind of symmetry simplifies the mathematical treatment of the surface.

The equation \(r = f(z)\) means for any given height \(z\), the value of \(r\) does not change as you rotate around the \(z\)-axis. This indicates that the surface retains a consistent shape as it spins around this central axis.
  • Imagine a spinning top or a round vase, where the shape does not change as you look around it.
  • Axisymmetry is fundamental in designs involving cylindrical shapes, like pipes and tunnels, ensuring uniform properties around an operational axis.
Rotational Symmetry
Rotational symmetry refers to the characteristic of an object or surface to look the same after a certain degree of rotation. In the context of the surface described by \(r = f(z)\), the rotational symmetry is about the \(z\)-axis.

Since the surface equation is independent of \(\theta\), rotating the object by any angle \(\theta\) will result in the same configuration. This means the object has infinite order rotational symmetry about its axis.
  • For instance, rotating a perfect circle by any angle will look unchanged, which is an essence of rotational symmetry.
  • In practical applications, rotational symmetry is utilized in objects like wheels and gears to ensure consistent functionality from all rotational positions.

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Most popular questions from this chapter

Approximate the double integral of \(f(x, y)\) over the region \(R\) partitioned by the given vertical lines \(x=a\) and horizontal lines \(y=c .\) In each subrectangle, use \(\left(x_{k}, y_{k}\right)\) as indicated for your approximation. $$\iint_{R} f(x, y) d A=\sum_{i=1}^{n} f\left(x_{k}, y_{k}\right) \Delta A_{k}$$ \(f(x, y)=x+2 y\) over the region \(R\) inside the circle \((x-2)^{2}+(y-3)^{2}=1\) using the partition \(x=1,3 / 2,2,5 / 2\) 3 and \(y=2,5 / 2,3,7 / 2,4\) with \(\left(x_{k}, y_{k}\right)\) the center (centroid) in the \(k\) th subrectangle (provided the subrectangle lies within \(R\) )

In polar coordinates, the average value of a function over a region \(R\) (Section 15.3 ) is given by $$\frac{1}{\operatorname{Area}(R)} \iint_{R} f(r, \theta) r d r d \theta$$ Find the average height of the hemispherical surface \(z=\sqrt{a^{2}-x^{2}-y^{2}}\) above the disk \(x^{2}+y^{2} \leq a^{2}\) in the \(x y\) -plane.

Evaluate the integral \(\iint_{R}\left(x^{2}+y^{2}\right)^{-2} d A,\) where \(R\) is the region inside the circle \(x^{2}+y^{2}=2\) for \(x \leq-1\)

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x$$

Evaluate the spherical coordinate integrals. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{\sec \phi}(\rho \cos \phi) \rho^{2} \sin \phi d \rho d \phi d \theta$$
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