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Coordinate Transformation Formula Sheet^1 Table with the Del operator in rectangular, cylindrical, and spherical coordinatesOperation Cartesian coordinates (x,y,z)^
Cylindrical coordinates(ρ, φ ,z)^
Spherical coordinates (r,θ,^ φ ) Definitionofcoordinates^ ˆ^ ˆ^
ˆcos sin φ φ= + ρ x y ˆ ˆ ˆsin cosφ φ φ= − + x y ˆ ˆ= z z Definitionofunitvectors^ ˆ^ ˆ^
ˆ^ ˆ^ ˆ ˆˆsin^ cos^ θ^ θ=^ + r^ ρ^ z sin cos sin sin cos θ φ θ φ θ= + + r x y z ˆ ˆ^ ˆ^ ˆ ˆ ˆcos cos cos sin sinθ φ θ φ θ= + − θ x y z ˆcos^ sinθ^ θ=^ − θ^ ρ^ z ˆ ˆ^ ˆˆ ˆsin cosφ φ φ= − + x y φ^ φ=
2 A vector field Gradient^ Divergence Curl^ Laplaceoperator^ VectorLaplacian^ Differentialdisplacement^ Differentialnormal area
4 Vector Transformations in Rectangular, Cylindrical, and Spherical Coordinates
From rectangular to cylindrical:^ cos^ sin^
0^ x sin cos 0^ y 0 0 1 z^
z A^
A
A^
A
A^
A
φ^ ρ φ
⎛^ ⎞^ ⎛^
⎞⎛^ ⎞
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
=^ −⎜ ⎟ ⎜^
⎟⎜^ ⎟
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
⎝^ ⎠^ ⎝^
⎠⎝^ ⎠
From cylindrical to rectangular:^ cos^ sin^
(^0) sin cos 0 0 0 1 x y z^
z A^
A
A^
A
A^
ρ φ A
φ^ φ−⎛ ⎞ ⎛ ⎞⎛ φ^ φ
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
=⎜ ⎟ ⎜^
⎟⎜^ ⎟
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
⎝^ ⎠^ ⎝^
⎠⎝^ ⎠
From rectangular to spherical: sin cos^ sin^ sin^ cos cos cos^ cos^ sin^ sin sin^ cos^05 r^
x y z A^
A
A^
A
A^
A
θ^ φ θ φ
θ^ φ^
θ^ φ^ θ^
φ^ θ φφ
⎛^ ⎞^ ⎛^
⎞⎛^ ⎞
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
=^
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
⎜^
⎟⎜^ ⎟
⎜^ ⎟^ −⎝^
⎠⎝^ ⎠
⎝^ ⎠^ From spherical to rectangular:^ sin^ cos
cos^ cos^
sin sin^ sin^ cos^
sin^ cos cos sin^0 x^
r A^ y z
A
A^
A
A^
θ A φ
θ^ φ^ θ^
φ^ φ
θ^ φ^ θ^
⎛^ φ φ θθ
⎛^ ⎞^ ⎛^
⎞ ⎜^ ⎟
⎜^ ⎟^ ⎜^
=^
⎜^ ⎟
⎜^ ⎟^ ⎜^
⎜^ ⎟^ ⎜^
⎟⎜^ ⎟
⎝^ ⎠^ ⎝^
⎠⎝^ ⎠
