Question

Determine whether the following statements are true and give an explanation or counterexample. a. Any point on the \(z\) -axis has more than one representation in both cylindrical and spherical coordinates. b. The sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) are the same.

Short answer

Are the sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) equal? Answer: Yes, every point on the z-axis has more than one representation in both cylindrical and spherical coordinates. The sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) are equal as they both represent the same cone with its vertex at the origin and opening upward along the z-axis.

Step-by-step solution

a. Statement analysis

We need to analyze the given statement: "Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates." First, let's convert the coordinates of a point on the z-axis to cylindrical and spherical coordinates. A point on the z-axis has the form \((0, 0, z)\). In cylindrical coordinates, we have: 1. \(r = \sqrt{x^2 + y^2} = \sqrt{0^2 + 0^2} = 0\) 2. \(\theta =\) any angle, as \(x\) and \(y\) are both zero 3. \(z=z\) Thus, any point on the z-axis can be represented as \((0, \theta, z)\) in cylindrical coordinates, where \(\theta\) can be any angle. This means that every point on the z-axis has more than one representation in cylindrical coordinates. In spherical coordinates, we have: 1. \(\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{0^2 + 0^2 + z^2} = |z|\) 2. \(\varphi = \mathrm{acos}(\frac{z}{\rho}) = \mathrm{acos}(\pm1)\) 3. \(\theta =\) any angle, as \(x\) and \(y\) are both zero Thus, any point on the z-axis can also be represented as \((|z|, \mathrm{acos}(\pm1), \theta)\) in spherical coordinates, where \(\theta\) can be any angle. This means that every point on the z-axis has more than one representation in spherical coordinates as well. So, the statement is true.

b. Comparison of sets

We need to check if the given sets are equal, i.e., if \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) represent the same set of points. First, let's analyze the set \(\\{(r, \theta, z): r=z\\}\). In this set, each point has coordinates \((r,\theta, z)\), and the condition \(r=z\) must hold. For a point in cylindrical coordinates, we know that \(r=\sqrt{x^2+y^2}\). Thus, we have \(\sqrt{x^2+y^2}=z\). This equation represents a cone with its vertex at the origin and opening upward along the z-axis. Now, let's analyze the set \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\). In this set, each point has coordinates \((\rho, \varphi, \theta)\), and the condition \(\varphi=\pi / 4\) must hold. For a point in spherical coordinates, we know that \(z=\rho\cos\varphi\) and \(r=\rho\sin\varphi\). Since \(\varphi=\pi/4\), we have \(\sin\varphi=\cos\varphi=\frac{1}{\sqrt{2}}\). Thus, we get \(r=\frac{\rho}{\sqrt{2}}\) and \(z=\frac{\rho}{\sqrt{2}}\). This implies \(r=z\), which represents the same cone as described above in cylindrical coordinates. Therefore, the sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) are the same, and the statement is true.

Key concepts

Cylindrical Coordinates

Cylindrical coordinates offer a nifty way to represent points in space by combining two-dimensional polar coordinates with a height element. Imagine a point located in a 3D space with Cartesian coordinates

• A point on the z-axis is characterized by x=0 and y=0.

We translate these into cylindrical coordinates with the following parameters:

• The radius, \( r \), which is computed by \( r = \sqrt{x^2 + y^2} \).
• The angle, \( \theta \), which can be any value since we are on the z-axis (where \( x = 0 \) and \( y = 0 \)).
• The height, \( z \), remains the same as in Cartesian coordinates.

Thus, any point on the z-axis can easily have multiple representations expressed as \((0, \theta, z)\) because the angle can vary without any restrictions. This is what makes cylindrical coordinates particularly interesting and useful for rotationally symmetric problems.

Spherical Coordinates

Spherical coordinates provide another fascinating approach to describing points in three-dimensional space. In this system, a point's position is determined not by its direct distance along Cartesian axes but by its radial distance and two angles.
For points on the z-axis, things simplify significantly:

• The radius, \( \rho \), describes the straight-line distance from the origin to the point and is expressed as \( \rho = \sqrt{x^2 + y^2 + z^2} \), which simplifies to \( |z| \).
• The inclination angle, \( \varphi \), is determined by the angle between the radius \( \rho \) and the positive z-axis - thus \( \varphi = \cos^{-1}\left(\frac{z}{\rho}\right) = \cos^{-1}(\pm 1) \). It is 0 or \( \pi \) since the point lies directly along the z-axis.
• The azimuthal angle, \( \theta \), can be any angle since the x and y coordinates are zero, allowing variability.

Hence, like cylindrical coordinates, spherical coordinates also allow numerous representations of the same point on the z-axis due to multiple possible values of \( \theta \).

Representation of Points

The representation of points in various coordinate systems gives unique perspectives and insights into geometry and physics problems. In cylindrical coordinates, a point is determined by a radius from the z-axis, an angle within the xy-plane, and height above or below that plane.

• For spherical points, you define its distance from the origin and two angles that specify its orientation.

Let's consider how these different methods of representation enable a more comprehensive understanding of spatial relationships. Using cylindrical and spherical coordinates, each point has multiple representations because angles like \( \theta \) can be varied without altering the ultimate location, especially noticeable on axes or symmetric points.

• This flexibility is profound when solving physics problems, such as computing the path of an object in rotational motion or integrating over a sphere.

Analyzing Statements

Analyzing mathematical statements often requires more than computation; it involves understanding the underlying concepts. Take for instance, the statement about the sets \((r, \theta, z): r=z\) in cylindrical coordinates and \((\rho, \varphi, \theta): \varphi=\pi/4\) in spherical.

• In cylindrical coordinates, \( r=z \) forms a specific relation known as a conical surface where the radius directly aligns with the vertical height at every point.
• In spherical terms, such a condition is mirrored when \( \, \varphi = \pi/4 \, \) meaning the same inclination applies globally, creating another conic shape.

By converting between cylindrical and spherical systems, the equivalency of these sets as geometric entities becomes apparent. This showcases how flexibility in representations confirms consistent geometrical truths across different frameworks.

• Such abilities are useful in advanced calculations and provide better visualization tools when tackling complex shapes and descriptions.