cylindrical polar coordinates

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Suppose that three mutually perpendicular directed lines Ox, Oy and Oz, intersecting at the point O and forming a right-handed system, are taken as coordinate axes. For any point P, let M and N be the projections of P onto the xy-plane and the z-axis respectively. Then ON=PM=z, the z-coordinate of P. Let ρ=| PN |, the distance of P from the z-axis, and let ϕ be the angle ∠ xOM in radians (0≤ϕ<2 π). Then (ρ, ϕ, z) are the cylindrical polar coordinates of P. (It should be noted that the points of the z- axis give no value for ϕ.) The two coordinates (ρ, ϕ) can be seen as polar coordinates of the point M and, as with polar coordinates, ϕ+2, where k is an integer, may be allowed in place of ϕ.The Cartesian coordinates (x, y, z) of P can be found from (ρ, ϕ, z) by: x=ρ cos ϕ, y=ρ sin ϕ, and z=z. Conversely, the cylindrical polar coordinates can be found from (x, y, z) by: is such that and and z=z. Cylindrical polar coordinates can be useful in treating problems involving right-circular cylinders. Such a cylinder with its axis along the z-axis then has equation ρ=constant.

Subjects: Mathematics.

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