Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.

tinfoQ 2021-02-05 Answered
Provide notes on how triple integrals defined in cylindrical and spherical coordinates and the reason to prefer one of these coordinate systems to working in rectangular coordinates.
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Gennenzip
Answered 2021-02-06 Author has 96 answers
Cylindrical coordinates prefer represent a point P in space by ordered triples (r,θ, z) in which r and θ are polar coordinates for the vertical projection of P on the xy=pla, with r  0, and x is the rectangular vertical coordinate.
The equations relating rectangular (x, y, z) and cylindrical (r,θ, z) coordinates are,
x=rcosθ
y=rsinθ
z=z
r2=x2 + y2
tanθ=yx
The spherical coordinates represent a point P in by ordered triples (ρ,ϕ,θ) in which,
ρ is the distance from P to the origin (ρ  0)
. ϕ is the angle over{OP} makes with the positive z - axis (0  ϕ  π)
. θ is the angle from cylindrical coordinates.
The equations relating spherical coordinates to Cartesian and cylindrical coordinates are,
r=ρsinϕ
x=rcosθ=ρsinϕcosθ
z=ρcosϕ
y=rsinθ=ρsinϕsinθ
ρ=x2 + y2 + z2=r2 + z2
tanθ=yx
Cylindrical coordinates are good for describing cylinders whose axes run along the z - axis and planes that either contain the z - axis or lie perpendicular to the z -axis.
Surfaces like these have equations of constant coordinate value.

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