Note
- This page uses standard physics notation. For spherical coordinates, is the angle between the z axis and the radius vector connecting the origin to the point in question. is the angle between the projection of the radius vector onto the x-y plane and the x axis. Some sources reverse the definitions of and, so the meaning should be inferred from the context.
- The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan(y/x) has an image of (-π/2, +π/2), whereas atan2(y, x) is defined to have an image of (-π, π]. (The expressions for the Del in spherical coordinates may need to be corrected)
Operation | Cartesian coordinates (x,y,z) | Cylindrical coordinates (ρ,φ,z) | Spherical coordinates (r,θ,φ) | Parabolic cylindrical coordinates (σ,τ,z) |
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Definition of coordinates |
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Definition of unit vectors |
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A vector field | ||||
Gradient | ![]() |
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Divergence | ![]() |
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Curl | ![]() |
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Laplace operator | ![]() |
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Vector Laplacian | ![]() |
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Material derivative
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Differential displacement | ||||
Differential normal area | ![]() |
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Differential volume | ||||
Non-trivial calculation rules:
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Read more about this topic: Del In Cylindrical And Spherical Coordinates
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