# Cylindrical / Spherical Coordinates

• eurekameh
In summary, the conversation revolves around converting a Cartesian coordinate system to cylindrical and spherical coordinate systems. The cylindrical system is represented by r,vector = er,hat + sint(e3,hat), while the spherical coordinate system is represented by r = sqrt(x^2+y^2+z^2) = sqrt(1+sin^2(t)). There is a question about simplifying the expression 1/cos[tan^-1*(sint)], and a discussion about r = sqrt(x^2+y^2) = 1 for cylindrical coordinates. The speaker also expresses interest in checking this in an analytic way.
eurekameh
I'm trying to convert the below Cartesian coordinate system into cylindrical and spherical coordinate systems. For the cylindrical system, I had r,vector = er,hat + sint(e3,hat).
While I do have a technically correct answer for the spherical coordinate system, I believe, I was wondering if there was a way to simplify the expression 1/cos[tan^-1*(sint)]. Thanks.

##r=\sqrt{x^2+y^2+z^2} = \sqrt{1+\sin^2(t)}##?

Also, am I right in thinking r = rad(x^2 + y^2) = 1 for cylindrical coordinates?

eurekameh said:
Would be interesting to check this in an analytic way.

Also, am I right in thinking r = rad(x^2 + y^2) = 1 for cylindrical coordinates?
If you use (r,theta,z) as coordinates: Right.

Hello,

Thank you for sharing your progress in converting Cartesian coordinates into cylindrical and spherical systems. It seems like you have a good understanding of the concepts so far.

To answer your question, there is a way to simplify the expression 1/cos[tan^-1(sint)]. This expression can be rewritten as sec[tan^-1(sint)]. Using the trigonometric identity sec^2x = 1 + tan^2x, we can simplify it further to 1 + (sint)^2, which can be represented as r^2 in spherical coordinates. Therefore, the simplified expression for the spherical coordinate system would be r,theta,phi = r, r,hat + rsin(theta),cos(phi),hat.

I hope this helps with your conversion process. Keep up the good work!

## What are cylindrical and spherical coordinates?

Cylindrical and spherical coordinates are two different coordinate systems used to describe points in three-dimensional space. Cylindrical coordinates use the distance from the origin, the angle from the positive x-axis, and the height from the xy-plane to describe a point. Spherical coordinates use the distance from the origin, the azimuth angle from the positive x-axis, and the polar angle from the positive z-axis to describe a point.

## When are cylindrical coordinates typically used?

Cylindrical coordinates are typically used when dealing with objects or situations that have cylindrical symmetry, such as cylinders, cones, and rotational motion.

## What is the conversion between cylindrical and spherical coordinates?

The conversion between cylindrical and spherical coordinates is as follows:
r = √(ρ^2 + z^2)
θ = arctan(y/x)
φ = arctan(√(x^2+y^2)/z)
where r is the distance from the origin, ρ is the distance from the xy-plane, θ is the angle from the positive x-axis, and φ is the angle from the positive z-axis.

## What are the advantages of using spherical coordinates?

Spherical coordinates are advantageous when dealing with objects or situations that have spherical symmetry, such as planets, stars, and spherical-shaped particles. They also simplify certain mathematical equations, making them easier to solve.

## How are cylindrical and spherical coordinates used in physics and engineering?

Cylindrical and spherical coordinates are commonly used in physics and engineering to describe the motion, position, and forces acting on objects in three-dimensional space. They are also used to solve various types of differential equations, such as the heat equation and wave equation, in these fields.

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