Jump to content

Coordinate systems/Derivation of formulas

From Wikiversity

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>
  2. Contribute to Wikiveristy by linking the title to a discussion and/or proof. Just click the redlink and start the page.

If you just came in from Wikipedia, the rules about treating each other with respect and obeying copyright laws remain more or less the same as on Wikipedia, but the definition of what constitutes an acceptable article is a lot looser. Here we call them "resources". Welcome to the wacky world of Wikiversity, where anything goes.

Transformations between coordinates

[edit | edit source]
  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)

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

  ,       ,      verified using mathworld[1]

  ,       ,      verified using mathworld[2]

  ,       ,      verified using mathworld[3]

  ,       ,      --no reference

  ,       ,      verified using mathworld[4]

  ,       ,      no reference

  ,       ,      no reference

  ,       ,      no reference

 Verified, see page linked in title

 Verified, see page linked in title

 Verified, see page linked in title

 Verified, see page linked in title

Vector and scalar fields

[edit | edit source]

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

is the w:gradient of a scalar field.

is the w:divergence of a vector field

is the w:curl (mathematics) of A

is the w:Laplace operator on a scalar field

is the w:Vector Laplacian of

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

Differential normal area

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

Non-trivial calculation rules:

  1. (Lagrange's formula for del)

References

[edit | edit source]


  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. Cite error: Invalid <ref> tag; no text was provided for refs named Mathworld
  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

[1]

[2]



  1. Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011.
  2. Huba J.D. (1994). "NRL Plasma Formulary revised" (PDF). Office of Naval Research. Retrieved 11 June 2014.


Backup copy from Wikipedia

[edit | edit source]

Copy or read but never change Original Copy from Wikipedia