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Χρήστης
:
Kupirijo/Συντεταγμένες
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Επεξεργασία
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Χρήστης:Kupirijo
Βαθμωτό πεδίο
Επεξεργασία
Operation
Καρτεσιανές συντεταγμένες
(x,y,z)
Κυλινδρικές συντεταγμένες
(s,φ,z)
Σφαιρικές συντεταγμένες
(r,θ,φ)
Ορισμός
συντεταγμένων
Καρτεσιανές συντεταγμένες
x
=
s
cos
ϕ
y
=
s
sin
ϕ
z
=
z
{\displaystyle {\begin{matrix}x&=&s\cos \phi \\y&=&s\sin \phi \\z&=&z\end{matrix}}}
x
=
r
sin
θ
cos
ϕ
y
=
r
sin
θ
sin
ϕ
z
=
r
cos
θ
{\displaystyle {\begin{matrix}x&=&r\sin \theta \cos \phi \\y&=&r\sin \theta \sin \phi \\z&=&r\cos \theta \end{matrix}}}
Κυλινδρικές συντεταγμένες
s
=
x
2
+
y
2
ϕ
=
arctan
(
y
/
x
)
z
=
z
{\displaystyle {\begin{matrix}s&=&{\sqrt {x^{2}+y^{2}}}\\\phi &=&\arctan(y/x)\\z&=&z\end{matrix}}}
s
=
r
sin
(
θ
)
ϕ
=
ϕ
z
=
r
cos
(
θ
)
{\displaystyle {\begin{matrix}s&=&r\sin(\theta )\\\phi &=&\phi \\z&=&r\cos(\theta )\end{matrix}}}
Σφαιρικές συντεταγμένες
r
=
x
2
+
y
2
+
z
2
θ
=
arctan
(
x
2
+
y
2
z
)
ϕ
=
arctan
(
y
/
x
)
{\displaystyle {\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}}}
r
=
s
2
+
z
2
θ
=
arctan
(
s
/
z
)
ϕ
=
ϕ
{\displaystyle {\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
s
^
=
x
s
x
^
+
y
s
y
^
ϕ
^
=
−
y
s
x
^
+
x
s
y
^
z
^
=
z
^
{\displaystyle {\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}}}
x
^
=
cos
ϕ
s
^
−
sin
ϕ
ϕ
^
y
^
=
sin
ϕ
s
^
+
cos
ϕ
ϕ
^
z
^
=
z
^
{\displaystyle {\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}}}
x
^
=
sin
θ
cos
ϕ
r
^
+
cos
θ
cos
ϕ
θ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
θ
sin
ϕ
r
^
+
cos
θ
sin
ϕ
θ
^
+
cos
ϕ
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\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}}}
σ
^
=
τ
τ
2
+
σ
2
x
^
−
σ
τ
2
+
σ
2
y
^
τ
^
=
σ
τ
2
+
σ
2
x
^
+
τ
τ
2
+
σ
2
y
^
z
^
=
z
^
{\displaystyle {\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}}}
r
^
=
x
x
^
+
y
y
^
+
z
z
^
r
θ
^
=
x
z
x
^
+
y
z
y
^
−
s
2
z
^
r
s
ϕ
^
=
−
y
x
^
+
x
y
^
s
{\displaystyle {\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}}}
r
^
=
s
r
s
^
+
z
r
z
^
θ
^
=
z
r
s
^
−
s
r
z
^
ϕ
^
=
ϕ
^
{\displaystyle {\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}}}
s
^
=
sin
θ
r
^
+
cos
θ
θ
^
ϕ
^
=
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\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}}}
{\displaystyle {\begin{matrix}\end{matrix}}}