Polar, Cylindrical, and Spherical Coordinates Calculator

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Example

Created on 2024-06-20Asked by William Adams (Solvelet student)

Convert the point

P(3, \pi/4, 2)

from cylindrical coordinates to spherical coordinates.

Solution

To convert the point

P(3, \pi/4, 2)

from cylindrical coordinates to spherical coordinates: 1. The conversion formulas are:

\begin{aligned} r &= \sqrt{\rho^2 + z^2}, \\ \theta &= \theta, \\ \phi &= \arctan\left(\frac{\rho}{z}\right). \end{aligned}

2. Substitute

\rho = 3

and

z = 2

into the conversion formulas:

\begin{aligned} r &= \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13}, \\ \theta &= \frac{\pi}{4}, \\ \phi &= \arctan\left(\frac{3}{2}\right). \end{aligned}

3. Calculate the value of

\phi

\phi \approx \arctan\left(\frac{3}{2}\right) \approx 0.982 \text{ radians}.

Therefore, the point

P(3, \pi/4, 2)

in spherical coordinates is approximately

P(\sqrt{13}, \frac{\pi}{4}, 0.982)

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Daniel King on Solvelet

1. Convert the point

(2, \frac{\pi}{4}, 3)

from cylindrical coordinates to spherical coordinates.2. Find the volume of the region enclosed by the sphere

\rho = 2\sin(\phi)

in spherical coordinates.

DefinitionPolar coordinates are based on a radius and an angle in a plane. Cylindrical coordinates add a height dimension to polar coordinates. Points in 3D space are exactly defined by their radius, or distance up off the x-y plane, their polar angle, and their azimuthal angle by using spherical coordinates. Example: A point (r,θ,z) with cylindrical coordinates is equal to (rcosθ,rsinθ,z) in Cartesian coordinates.

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