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Chapter 16: Problem 2

Explain how spherical coordinates are used to describe a point in \(\mathrm{R}^{3}\).

Short Answer

Expert verified
Based on the given information about spherical coordinates, provide a short answer explaining how this coordinate system can be used to describe a point in \(\mathrm{R}^{3}\). Spherical coordinates provide an alternative way to describe a point in three-dimensional space using three parameters: radial distance (\(\rho\)), polar angle (\(\theta\)), and azimuthal angle (\(\phi\)). This system is particularly useful in situations involving symmetry or distances from a central point. A point in Cartesian coordinates can be converted to spherical coordinates using specific conversion formulas, and vice versa, allowing for flexibility in mathematical analysis and problem-solving.

Step by step solution

01

Introduction to spherical coordinates

Spherical coordinates system is an alternative coordinate system in 3-dimensional spaces. It represents a point in \(\mathrm{R}^{3}\) using three parameters: the radial distance, the polar angle, and the azimuthal angle. These parameters are represented by \((\rho, \theta, \phi)\), where - \(\rho \geq 0\): the radial distance from the origin to the point of interest. - \(\theta\): the polar angle, measured from the positive z-axis, to the point of interest. It ranges between \(0 \leq \theta \leq \pi\). - \(\phi\): the azimuthal angle, measured from the positive x-axis to the projection of the point of interest onto the xy-plane. It ranges between \(0 \leq \phi < 2\pi\).
02

Conversion from Cartesian coordinates to spherical coordinates

Given a point \((x, y, z)\) in Cartesian coordinates, we can convert it into spherical coordinates \((\rho, \theta, \phi)\) with the following formulas: 1. \(\rho = \sqrt{x^2 + y^2 + z^2}\) 2. \(\theta = \arccos\left(\frac{z}{\rho}\right)\) 3. \(\phi = \arctan2(y, x)\), where \(\arctan2\) returns a value in the range of \([0, 2\pi)\) based on the values of y and x.
03

Conversion from spherical coordinates to Cartesian coordinates

Given a point \((\rho, \theta, \phi)\) in spherical coordinates, we can convert it into Cartesian coordinates \((x, y, z)\) with the following formulas: 1. \(x = \rho\sin(\theta)\cos(\phi)\) 2. \(y = \rho\sin(\theta)\sin(\phi)\) 3. \(z = \rho\cos(\theta)\)
04

Example of a point conversion between Cartesian and spherical coordinates

Let's take a point \(P(3, 4, 5)\) in Cartesian coordinates. We will now convert it into spherical coordinates: 1. Calculate the radial distance, \(\rho\): \(\rho = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50}\) 2. Calculate the polar angle, \(\theta\): \(\theta = \arccos\left(\frac{5}{\sqrt{50}}\right) \approx 0.6435 \; \text{radians}\) 3. Calculate the azimuthal angle, \(\phi\): \(\phi = \arctan2(4, 3) \approx 0.9273 \; \text{radians}\) Now, we have the spherical coordinates for point \(P\): \((\rho, \theta, \phi) = (\sqrt{50}, 0.6435, 0.9273)\). To validate that our conversion is correct, let's convert these spherical coordinates back to Cartesian coordinates: 1. Calculate the x-coordinate: \(x = \sqrt{50}\sin(0.6435)\cos(0.9273) \approx 3\) 2. Calculate the y-coordinate: \(y = \sqrt{50}\sin(0.6435)\sin(0.9273) \approx 4\) 3. Calculate the z-coordinate: \(z = \sqrt{50}\cos(0.6435) \approx 5\) As the converted Cartesian coordinates \((3, 4, 5)\) match the original point, the conversion between Cartesian and spherical coordinates was successful.

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Key Concepts

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

Radial Distance
In spherical coordinates, the radial distance is denoted by \( \rho \). It measures how far a point is from the origin in 3-dimensional space. Picture it as the length of a straight line, extending directly from the origin to the point in question. This distance is always non-negative (\( \rho \geq 0 \)).
It acts much like the radius of a sphere centered at the origin, hence the name 'radial'. Imagine every point at the same radial distance defines a sphere of that radius. It's calculated using the formula:
\[ \rho = \sqrt{x^2 + y^2 + z^2} \]
where \( x \), \( y \), and \( z \) are the Cartesian coordinates of the point.
Understanding radial distance helps in visualizing spherical coordinates, making it easier to switch between different systems to describe a point.
Polar Angle
The polar angle, represented as \( \theta \), is another crucial component of spherical coordinates. It measures the angle between the positive z-axis and the line connecting the origin to the point. This angle is typically measured in radians, ranging from 0 to \( \pi \) (180 degrees).
Visualize it like a vertical clock hand sweeping from top (at \( z \)-axis) toward the xy-plane. The angle tells you how 'tilted' a point is from the z-axis. You can calculate it with:
\[ \theta = \arccos\left(\frac{z}{\rho}\right) \]
This angle helps in determining the vertical position of a point in 3-dimensional space.
This concept is vital for converting back and forth between coordinate systems, enhancing your geometric intuition.
Azimuthal Angle
The azimuthal angle, denoted by \( \phi \), complements the polar angle by describing a point's rotational position around the z-axis. Think of it like how longitude works on a globe, guiding you to the correct point on a horizontal circle.
\( \phi \) can take any value within the range [0, \( 2\pi \)], allowing complete rotation from zero to a full circle. It's calculated using:
\[ \phi = \arctan2(y, x) \]
where \( \arctan2 \) helps determine the angle based on the x and y coordinates, ensuring the correct quadrant is identified. This angle represents the point's projection onto the xy-plane, rotating around the z-axis, which is crucial for understanding spherical coordinates in three dimensions.
Cartesian Coordinates
Cartesian coordinates, represented as (x, y, z), are a familiar way of describing positions in 3-dimensional space using perpendicular axes. In this system, each coordinate corresponds to the point's distance along each axis:
  • \( x \) for the horizontal position
  • \( y \) for the vertical position on the xy-plane
  • \( z \) for the height above or below the xy-plane
Understanding Cartesian coordinates is the foundation for learning other systems, like spherical coordinates. They provide clarity when visualizing or solving geometric problems in space. Additionally, Cartesian coordinates simplify the process of transitioning to spherical coordinates through formulas, helping you appreciate different perspectives of the same point.
Coordinate Conversion
Converting between Cartesian and spherical coordinates involves using precise mathematical formulas to transition between two perspectives of space. This conversion is essential for tasks requiring different vantage points, such as physics problems or computer graphics.

From Cartesian to spherical, you have:
  • Radial distance: \( \rho = \sqrt{x^2 + y^2 + z^2} \)
  • Polar angle: \( \theta = \arccos\left(\frac{z}{\rho}\right) \)
  • Azimuthal angle: \( \phi = \arctan2(y, x) \)
From spherical to Cartesian, the conversions are:
  • \( x = \rho\sin(\theta)\cos(\phi) \)
  • \( y = \rho\sin(\theta)\sin(\phi) \)
  • \( z = \rho\cos(\theta) \)
The conversion process expands your understanding of space, enabling the use of both coordinate types effectively based on problem requirements.

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Most popular questions from this chapter

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by one leaf of the rose \(r=\sin 2 \theta,\) for \(0 \leq \theta \leq \pi / 2\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by the cardioid \(r=3-3 \cos \theta\)

Write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume \(g\) is continuous on the region. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 2} \int_{0}^{4 \sec \varphi} g(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta\) in the orders \(d \rho d \theta d \varphi\) and \(d \theta\) d\rho \(d \varphi\).

Charge distribution A spherical cloud of electric charge has a known charge density \(Q(\rho),\) where \(\rho\) is the spherical coordinate. Find the total charge in the cloud in the following cases. a. \(Q(\rho)=\frac{2 \times 10^{-4}}{\rho^{4}}, 1 \leq \rho<\infty\). b. \(Q(\rho)=\left(2 \times 10^{-4}\right) e^{-0.01 p^{3}}, 0 \leq \rho<\infty\).

Sketch the following regions \(R\). Then express \(\iint_{R} g(r, \theta) d A\) as an iterated integral over \(R\) in polar coordinates. The region outside the circle \(r=1 / 2\) and inside the cardioid \(r=1+\cos \theta\)
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