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Chapter 11: Problem 62

Convert the point from cylindrical coordinates to spherical coordinates. \((-4, \pi / 3,4)\)

Short Answer

Expert verified
The spherical coordinates of the point \((-4, \pi / 3, 4)\) are \((\sqrt{32}, 3\pi / 4, \pi / 3)\).

Step by step solution

01

Identify Given Cylindrical Coordinates

The problem gives us the point in cylindrical coordinates as \((-4, \pi / 3, 4)\). Hence, \(r is -4\), \(\theta is \pi / 3\) and \(z is 4\).
02

Convert to Spherical Coordinates

We first calculate the value of \(\rho\): \(\rho = \sqrt{r^2 + z^2}\) Substitute \(r = -4\) and \(z = 4\) into the formula, we get \(\rho = \sqrt{(-4)^2 + (4)^2} = \sqrt{32}\). Next, we calculate the value of \(\phi\): \(\phi = \text{arctan} \frac{r}{z}\)Substitute \(r = -4\) and \(z = 4\) into the formula, \(\phi = \text{arctan} \frac{-4}{4} = \text{arctan} -1 = -\pi / 4\). But \(\phi\) should be in the interval \([0, \pi]\), so we add \(\pi\) to our result and get \(\pi - \pi / 4 = 3\pi / 4\). Finally, \(\theta\) remains the same in both coordinate systems:\(\theta = \pi / 3\).
03

Present Final Spherical Coordinates

After the transformations, the point in spherical coordinates is \((\sqrt{32}, 3\pi / 4, \pi / 3)\).

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Key Concepts

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

Cylindrical Coordinates
Cylindrical coordinates are an extension of polar coordinates into three dimensions, often used when dealing with problems involving symmetry around an axis. Instead of the typical (x, y, z) of Cartesian coordinates, cylindrical coordinates are given as \( (r, \theta, z) \). Here's what each part means:
  • \( r \): The radial distance from the z-axis. In the exercise, this is \-4, but we consider the absolute value for calculations.
  • \( \theta \): The angle in the xy-plane from the positive x-axis. Given as \((\pi / 3)\), it defines the direction in the plane.
  • \( z \): The height above the xy-plane. Here, it鈥檚 given as 4.
Cylindrical coordinates are very useful when working with problems involving circles, cylinders, and other circular objects. They provide a more intuitive understanding of rotational symmetry, making certain problems easier to solve.
Spherical Coordinates
Spherical coordinates provide another way to express points in 3D space, ideal for problems involving spheres and similar shapes. They're defined by \( (\rho, \phi, \theta) \):
  • \( \rho \): The distance from the origin to the point. It鈥檚 calculated using \( \sqrt{r^2 + z^2} \). In the exercise, \( \rho = \sqrt{32} \).
  • \( \phi \): The angle from the positive z-axis. Initially, we found \( \phi = -\pi/4 \), but it's corrected to fit the interval \( [0, \pi]\) by adding \( \pi, \) resulting in \( 3\pi/4 \).
  • \( \theta \): This remains the same as the cylindrical coordinate angle \( \theta \), which is \( \pi/3 \).
Spherical coordinates are particularly useful in fields like physics and engineering, where objects are often considered in relation to an origin point, such as planets, stars, or fields emanating from a central source.
Trigonometric Transformations
Trigonometric transformations are essential when converting between different coordinate systems. They relate angles and distances in one system to another, often through the use of basic trigonometric functions like sine, cosine, and tangent.

In the coordinate conversion process, we used:
  • For calculating \( \rho \), we applied Pythagoras鈥� theorem, taking the square root of the sum of squares of \( r \) and \( z \).
  • The tangent function helped determine \( \phi \) by finding \( \text{arctan} (r/z) \). Adjustments may be needed to ensure the angle fits the desired range.
  • Being comfortable with these transformations allows seamless navigation between different systems.
Mastering trigonometric transformations opens up new ways to approach complex mathematical problems, providing a deeper insight into spatial relationships.

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Most popular questions from this chapter

\mathrm{\\{} P r o g r a m m i n g ~ Given vectors \(\mathbf{u}\) and \(\mathbf{v}\) in component form, write a program for a graphing utility in which the output is the component form of the projection of \(\mathbf{u}\) onto \(\mathbf{v}\).

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