   Chapter 11.7, Problem 81E

Chapter
Section
Textbook Problem

# Rectangular and Spherical Coordinates Give the equation for the coordinate conversion from rectangular to spherical coordinates and vice versa.

To determine

The equations for the co-ordinates conversion form rectangular co-ordinate to spherical co-ordinates and vice-versa.

Explanation

Takespherical co-ordinates as (ρ,θ,ϕ) and Rectangular co-ordinates as (x,y,z).

In a spherical co-ordinate system, a point P in space is represented by an ordered triple (r,θ,z) Where, ρ is the distance between P and the origin,ρ0 and ϕ is the angle between the positive z-axis and the line segment OP0ϕπ.

As shown figure below:

Rectangular to spherical co-ordinate conversion,

Consider the equationto calculate first co-ordinate

r=x2+y2=ρsinϕ

Use the Pythagorean Theorem to calculate first co-ordinate which states that the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base.

This can be written as, a2=b2+c2 and known as Pythagorean equation.

As shown in figure:

So,

ρ=r2+z2=x2+y2+z2

Hence,

ρ2=x2+y2+z2

Use trigonometric functionto calculate second co-ordinate

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