Χρήστης:Kupirijo/Συντεταγμένες

Βαθμωτό πεδίο

Operation		Καρτεσιανές συντεταγμένες (x,y,z)	Κυλινδρικές συντεταγμένες (s,φ,z)	Σφαιρικές συντεταγμένες (r,θ,φ)
Ορισμός συντεταγμένων	Καρτεσιανές συντεταγμένες		${\begin{matrix}x&=&s\cos \phi \\y&=&s\sin \phi \\z&=&z\end{matrix}}$	${\begin{matrix}x&=&r\sin \theta \cos \phi \\y&=&r\sin \theta \sin \phi \\z&=&r\cos \theta \end{matrix}}$
	Κυλινδρικές συντεταγμένες	${\begin{matrix}s&=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{matrix}}$		${\begin{matrix}s&=&r\sin(\theta )\\\phi &=&\phi \\z&=&r\cos(\theta )\end{matrix}}$
	Σφαιρικές συντεταγμένες	${\begin{matrix}r&=&{\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=&\arctan {\left({\frac {\sqrt {x^{2}+y^{2}}}{z}}\right)}\\\phi &=&\arctan(y/x)\\\end{matrix}}$	${\begin{matrix}r&=&{\sqrt {s^{2}+z^{2}}}\\\theta &=&\arctan {(s/z)}\\\phi &=&\phi \end{matrix}}$

Operation	Cartesian coordinates (x,y,z)	Cylindrical coordinates (s,φ,z)	Spherical coordinates (r,θ,φ)	Parabolic cylindrical coordinates (ο,τ,z)
Definition of unit vectors	${\begin{matrix}{\boldsymbol {\hat {s}}}&=&{\frac {x}{s}}\mathbf {\hat {x}} +{\frac {y}{s}}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\phi }}}&=&-{\frac {y}{s}}\mathbf {\hat {x}} +{\frac {x}{s}}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathbf {\hat {x}} &=&\cos \phi {\boldsymbol {\hat {s}}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \phi {\boldsymbol {\hat {s}}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$	${\begin{matrix}\mathbf {\hat {x}} &=&\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=&\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {\sigma }}}&=&{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} -{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\tau }}}&=&{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} +{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=&\mathbf {\hat {z}} \end{matrix}}$
	${\begin{matrix}\mathbf {\hat {r}} &=&{\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} +z\mathbf {\hat {z}} }{r}}\\{\boldsymbol {\hat {\theta }}}&=&{\frac {xz\mathbf {\hat {x}} +yz\mathbf {\hat {y}} -s^{2}\mathbf {\hat {z}} }{rs}}\\{\boldsymbol {\hat {\phi }}}&=&{\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{s}}\end{matrix}}$	${\begin{matrix}\mathbf {\hat {r}} &=&{\frac {s}{r}}{\boldsymbol {\hat {s}}}+{\frac {z}{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&=&{\frac {z}{r}}{\boldsymbol {\hat {s}}}-{\frac {s}{r}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\end{matrix}}$	${\begin{matrix}{\boldsymbol {\hat {s}}}&=&\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&=&{\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=&\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\\\end{matrix}}$	${\begin{matrix}\end{matrix}}$