Coordinate systems/Derivation of formulas

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The purpose of this resource is to carefully examine the Wikipedia article Del in cylindrical and spherical coordinates for accuracy.

The identities are reproduced below, and contributors are encouraged to either:

  1. Verify the identity and place its reference using a five em padding after the equation: {{pad|5em}}verified<ref>reference</ref>
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Transformations between coordinates[edit]

  1. w:Cartesian coordinates (x, y, z)
  2. w:Cylindrical coordinates (ρ, ϕ, z)
  3. w:Spherical coordinates (r, θ, ϕ)
  4. w:Parabolic cylindrical coordinates (σ, τ, z)

Coordinate variable transformations*[edit]

*Asterisk indicates that the title is a link to more discussion

Cylindrical from Cartesian variable transformation[edit]

  ,       ,      verified using mathworld[1]

Cartesian from cylindrical variable transformation[edit]

  ,       ,      verified using mathworld[2]

Cartesian from spherical variable transformation[edit]

  ,       ,      verified using mathworld[3]

Cartesian from parabolic cylindrical variable transformation[edit]

  ,       ,      --no reference

Spherical from Cartesian variable transformation[edit]

  ,       ,      verified using mathworld[4]

Spherical from cylindrical variable transformation[edit]

  ,       ,      no reference

Cylindrical from spherical variable transformation[edit]

  ,       ,      no reference

Cylindrical from parabolic cylindrical variable transformation[edit]

  ,       ,      no reference

Unit vectors[edit]

Cylindrical from Cartesian unit vectors[edit]

 Verified, see page linked in title

Cartesian from cylindrical unit vectors[edit]

 Verified, see page linked in title

Cartesian from spherical unit vectors[edit]

 Verified, see page linked in title

Parabolic cylindrical from Cartesian unit vectors[edit]

Spherical from Cartesian unit vectors[edit]

 Verified, see page linked in title

Spherical from cylindrical unit vectors[edit]

Cylindrical from spherical unit vectors[edit]

Vector and scalar fields[edit]

is vector field and f is a scalar field. The vector field can be expressed as:

Gradient of a scalar field[edit]

is the w:gradient of a scaler field.

Divergence of a vector field*[edit]

is the w:divergence of a vector field

Curl of a vector field[edit]

is the w:curl (mathematics) of A

Laplacian of a scalar field[edit]

is the w:Laplace operator on a scalar field

Laplacian of a vector field[edit]

is the w:Vector Laplacian of

Material derivative of a vector field[edit]

might be called the "convective derivative of B along A" (appropriate description if A' is a unit vector) [5]

Differential displacement[edit]

Differential normal areas[edit]

Differential normal area

Differential volume[edit]

  1.  verified[6]
  2.  verified[7]
  3.  verified[8]

nabla's on nabla's[edit]

Non-trivial calculation rules:

  1. (Lagrange's formula for del)

References[edit]


<references> [5]

[9]



  1. http://mathworld.wolfram.com/CylindricalCoordinates.html
  2. http://mathworld.wolfram.com/CylindricalCoordinates.html
  3. http://mathworld.wolfram.com/SphericalCoordinates.html
  4. http://mathworld.wolfram.com/SphericalCoordinates.html
  5. 5.0 5.1 Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. 
  6. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
  7. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
  8. James Stewart, Calculus: Concepts and Contexts, fourth edition, Brooks Cole 2005 pp. 884-5
  9. Huba J.D. (1994). "NRL Plasma Formulary revised". Office of Naval Research. Retrieved 11 June 2014. 

Backup copy from Wikipedia[edit]

Copy or read but never change Original Copy from Wikipedia