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Chapter 11: Problem 95

Give the equations for the coordinate conversion from rectangular to spherical coordinates and vice versa.

Short Answer

Expert verified
The formulas for converting from rectangular to spherical coordinates are: \( r = \sqrt{x^2 + y^2 + z^2} \), \( 胃 = \arctan(\frac{y}{x}) \), \( 蠁 = \arccos(\frac{z}{r}) \). On the other hand, the formulas for converting from spherical to rectangular coordinates are: \( x = r \sin(蠁) \cos(胃) \), \( y = r \sin(蠁) \sin(胃) \), \( z = r \cos(蠁) \).

Step by step solution

01

Rectangular to Spherical

The conversion from rectangular coordinates (x, y, z) to spherical coordinates (r, 胃, 蠁) is realized by the following formulas:\[r = \sqrt{x^2 + y^2 + z^2}\]\[胃 = \arctan\left(\frac{y}{x}\right)\]\[蠁 = \arccos\left(\frac{z}{r}\right)\]Here, \(r\) is the distance from the origin to the point, 胃 is the angle measured from the positive x-axis (counterclockwise), and 蠁 is the angle from the z-axis to the point.
02

Spherical to Rectangular

The conversion from spherical coordinates (r, 胃, 蠁) to rectangular coordinates (x, y, z) is realized by the following formulas:\[x = r \sin(蠁) \cos(胃)\]\[y = r \sin(蠁) \sin(胃)\]\[z = r \cos(蠁)\]Here, \(r\) is the distance from the origin to the point, 胃 is the angle measured from the positive x-axis (counterclockwise), and 蠁 is the angle from the z-axis to the point.

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