# Cylindrical polar co-ordinates

• Nylex
In summary, cylindrical polar coordinates are a system of representing points in three-dimensional space using a combination of distance from the origin, angle from a reference axis, and height above the xy-plane. They use a different set of variables compared to Cartesian coordinates and have a special relationship with spherical coordinates. To convert between Cartesian and cylindrical polar coordinates, specific formulas can be used. Cylindrical polar coordinates have various practical applications in engineering, physics, and computer graphics.
Nylex
My notes have an example of verifying the divergence theorem using cylindrical polars.

There's a vector field, A(r) = x(x-hat) + y(y-hat) + z^2(z-hat) and my notes say:

"Note that rho-hat = cos phi(x-hat) + sin phi(y-hat) and phi-hat = -sin phi(x-hat) + cos phi(y-hat) and so

A(r) = rho(rho-hat) + z^2(z-hat)".

Why?? I can't see any obvious connection between phi-hat, rho-hat and x and y .

## 1. What are cylindrical polar coordinates?

Cylindrical polar coordinates are a system of representing points in three-dimensional space using a combination of distance from the origin, angle from a reference axis, and height above the xy-plane.

## 2. How are cylindrical polar coordinates different from Cartesian coordinates?

Cylindrical polar coordinates use a different set of variables to represent points in three-dimensional space compared to Cartesian coordinates. While Cartesian coordinates use x, y, and z coordinates, cylindrical polar coordinates use r, θ, and z coordinates.

## 3. What is the relationship between cylindrical polar coordinates and spherical coordinates?

Cylindrical polar coordinates are a special case of spherical coordinates, where the angle θ is fixed at 90 degrees. This means that the z-coordinate in cylindrical polar coordinates is equivalent to the ρ-coordinate in spherical coordinates.

## 4. How do you convert between Cartesian and cylindrical polar coordinates?

To convert from Cartesian coordinates (x, y, z) to cylindrical polar coordinates (r, θ, z), you can use the following formulas: r = √(x² + y²), θ = arctan(y/x), and z = z.

## 5. What are some practical applications of cylindrical polar coordinates?

Cylindrical polar coordinates are often used in engineering and physics calculations, such as in fluid dynamics, electromagnetism, and heat transfer. They are also commonly used in computer graphics and 3D modeling to represent and manipulate three-dimensional objects.

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