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Can someone, please, show me an example of when you are better of with parabolic cylindrical coordinates than with cartesian coordinates when computing a triple integral over a solid?
FrogPad said:Things with symmetry around an axis, like a cylinder. Try this one with cartesian coordinates, then try it with cylindrical coordinates.
[tex] \int_{-2}^{2} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{2} \,\,\,(x^2+y^2)\,\,dz\,dy\,dx [/tex]
Parabolic cylindrical coordinates are a coordinate system used in mathematics and physics to describe points in a three-dimensional space. They consist of two parameters, ρ and θ, which represent the distance from the origin and the angle of rotation, respectively. This coordinate system is often used in problems involving cylindrical symmetry.
Unlike other coordinate systems, such as Cartesian or spherical coordinates, parabolic cylindrical coordinates have a parabolic instead of a circular cross-section. This allows for a more convenient representation of problems with cylindrical symmetry, such as heat conduction in a cylindrical object.
One of the main advantages of parabolic cylindrical coordinates is their ability to simplify mathematical equations in problems with cylindrical symmetry. They can also help in visualizing and understanding the behavior of functions in three-dimensional space.
Parabolic cylindrical coordinates can be transformed into other coordinate systems, such as Cartesian or spherical coordinates, using mathematical equations. This allows for easier integration and solving of problems involving multiple coordinate systems.
Parabolic cylindrical coordinates have numerous applications in physics and engineering, such as in solving problems related to heat conduction, fluid mechanics, and electromagnetic fields. They are also used in computer graphics and image processing for their ability to represent curved surfaces.