Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Change the following from spherical to Cartesian coordinates. (a) \((8, \pi / 4, \pi / 6)\) (b) \((4, \pi / 3,3 \pi / 4)\)

Step-by-Step Solution

Answer

(a) (2√2, 2√2, 4√3); (b) (√2, √6, -2√2).

Step 1: Understanding Spherical Coordinates

Spherical coordinates are given as \( (r, \theta, \phi) \), where: - \( r \) is the radial distance,- \( \theta \) is the azimuthal angle (angle from the x-axis in the xy-plane),- \( \phi \) is the polar angle (angle from the positive z-axis).Our task is to convert these spherical coordinates into Cartesian coordinates \((x, y, z)\). The formulas for conversion are:- \( x = r \sin \phi \cos \theta \)- \( y = r \sin \phi \sin \theta \)- \( z = r \cos \phi \)

Step 2: Spherical to Cartesian Conversion for (a)

Given spherical coordinates \((r, \theta, \phi) = (8, \pi/4, \pi/6)\):1. Calculate \( x \):\[ x = 8 \sin(\pi/6) \cos(\pi/4) = 8 \times \frac{1}{2} \times \frac{\sqrt{2}}{2} = 8 \times \frac{\sqrt{2}}{4} = 2\sqrt{2}\]2. Calculate \( y \):\[ y = 8 \sin(\pi/6) \sin(\pi/4) = 8 \times \frac{1}{2} \times \frac{\sqrt{2}}{2} = 8 \times \frac{\sqrt{2}}{4} = 2\sqrt{2}\]3. Calculate \( z \):\[ z = 8 \cos(\pi/6) = 8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}\]The Cartesian coordinates for (a) are \((2\sqrt{2}, 2\sqrt{2}, 4\sqrt{3})\).

Step 3: Spherical to Cartesian Conversion for (b)

Given spherical coordinates \((r, \theta, \phi) = (4, \pi/3, 3\pi/4)\):1. Calculate \( x \):\[ x = 4 \sin(3\pi/4) \cos(\pi/3) = 4 \times \frac{\sqrt{2}}{2} \times \frac{1}{2} = 4 \times \frac{\sqrt{2}}{4} = \sqrt{2}\]2. Calculate \( y \):\[ y = 4 \sin(3\pi/4) \sin(\pi/3) = 4 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = 4 \times \frac{\sqrt{6}}{4} = \sqrt{6}\]3. Calculate \( z \):\[ z = 4 \cos(3\pi/4) = 4 \times \left(-\frac{\sqrt{2}}{2}\right) = -2\sqrt{2}\]The Cartesian coordinates for (b) are \((\sqrt{2}, \sqrt{6}, -2\sqrt{2})\).

Key Concepts

Coordinate Transformation

The transformation from spherical to Cartesian coordinates involves converting three-dimensional points from the spherical system to the Cartesian system. In a spherical coordinate system, we represent a point in space using three values: radial distance (\(r\)), azimuthal angle (\(\theta\)), and polar angle (\(\phi\)). These are distinct from Cartesian coordinates (\(x, y, z\)), which directly measure distances along the three axes in a rectangular grid.

To perform this coordinate transformation, we use a set of mathematical formulas:

• For the x-coordinate: \(x = r \sin \phi \cos \theta\)
• For the y-coordinate: \(y = r \sin \phi \sin \theta\)
• For the z-coordinate: \(z = r \cos \phi\)

These formulas allow us to project the radial distance and angles into the Cartesian frame. You need both trigonometry and an understanding of spatial orientation to carry out these transformations effectively. Transforming coordinates is a fundamental skill in physics and engineering, as it allows for the analysis of problems in different coordinate systems.

Radial Distance

Radial distance is denoted by \(r\) in spherical coordinates and represents the straight-line distance from the origin to the point. Think of it as the "radius" of a sphere centered at the origin. The radial distance is always non-negative and gives us the magnitude of the position vector pointing from the origin to the point in space.

In physics, radial distance is crucial, particularly when dealing with gravitational or electric fields, where distance from the source affects the strength of the field. When converting between coordinate systems, this value remains constant and serves as the scaling factor for the transformation formulas. For example, in converting spherical to Cartesian coordinates, \(r\) acts as the scalar multiplier of the trigonometric functions involved in calculating x, y, and z.

Azimuthal Angle

The azimuthal angle, symbolized by \(\theta\), is the angle from the positive x-axis within the xy-plane. Imagine looking directly down at the point from above; \(\theta\) tells you how far around the circle you've moved from the x-axis, measured counterclockwise.

This angle ranges from 0 to \(2\pi\) radians (or 0 to 360 degrees) and is crucial for pinpointing a location's orientation in the horizontal plane. In spherical to Cartesian conversion, \(\theta\) helps determine the projection of the point onto the xy-plane. It's used in the conversion formula as \(\cos\theta\) and \(\sin\theta\), influencing the x and y coordinates, respectively.

Polar Angle

The polar angle, noted as \(\phi\), is measured from the positive z-axis toward the point in question. Unlike the azimuthal angle, \(\phi\) defines the altitude of the point relative to the z-axis. It ranges from 0 at the north pole of the sphere to \(\pi\) at the south pole, providing a vertical measure that tells you how far down from the zenith the point lies.

In spherical to Cartesian conversion, \(\phi\) critically affects all three Cartesian coordinates. Specifically, it helps to compute:

• The x and y projections by modifying the radius with \(\sin\phi\)
• The z-axis height by applying \(\cos\phi\)

By understanding \(\phi\), you can better visualize how high or low a point sits relative to the horizontal plane. Adjusting this angle is akin to angling a line down or up from a peak directly above the origin.