Silindirik ve küresel koordinatlarda del
Not

| işlem | Kartezyen koordinatlar (x, y,z) | Silindirik koordinatlar (ρ,φ,z) | Küresel koordinatlar (r,θ,φ) | Parabolik silindrik koordinatlar (σ,τ,z)   |
| --- | --- | --- | --- | --- |
| Koordinat Tanımları |  {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}} |  {\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}} |  {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}} |  {\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}} |
| Koordinat Tanımları |  {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}} |  {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}} |  {\begin{aligned}\rho &=r\sin(\theta )\\\phi &=\phi \\z&=r\cos(\theta )\end{aligned}} |  {\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}  |
| Birim Vektölerin Tanımları |  {\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}} |  {\begin{aligned}\mathbf {\hat {x}} &=\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}} |  {\begin{aligned}\mathbf {\hat {x}} &=\cos \phi \left(\sin \theta {\boldsymbol {\hat {r}}}+\cos \theta {\boldsymbol {\hat {\theta }}}\right)-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin \phi \left(\sin \theta {\boldsymbol {\hat {r}}}+\ cos\theta {\boldsymbol {\hat {\theta }}}\right)+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}} |  {\begin{aligned}{\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{aligned}} |
| Birim Vektölerin Tanımları |  {\begin{aligned}\mathbf {\hat {r}} &={\frac {x\mathbf {\hat {x}} \!+\!y\mathbf {\hat {y}} \!+\!z\mathbf {\hat {z}} }{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {z\left(x\mathbf {\hat {x}} \!+\!y\mathbf {\hat {y}} \right)\!-\!\left(x^{2}+y^{2}\right)\mathbf {\hat {z}} }{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\end{aligned}} |  {\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho }{\sqrt {\rho ^{2}+z^{2}}}}{\boldsymbol {\hat {\rho }}}+{\frac {z}{\sqrt {\rho ^{2}+z^{2}}}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&={\frac {z}{\sqrt {\rho ^{2}+z^{2}}}}{\boldsymbol {\hat {\rho }}}-{\frac {\rho }{\sqrt {\rho ^{2}+z^{2}}}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}} |  {\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\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{aligned}} |  {\begin{matrix}\end{matrix}}  |
| Bir vektör alanı \mathbf {A}  |  A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}}  |  A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}} |  A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}} |  A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }{\boldsymbol {\hat {z}}}  |
| Gradyan  \nabla f |  {\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}}  |  {\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}} |  {\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}} |  {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\boldsymbol {\hat {\tau }}}+{\partial f \over \partial z}{\boldsymbol {\hat {z}}}  |
| Diverjans  \nabla \cdot \mathbf {A}  |  {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z} |  {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z} |  {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial  \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi } |  {\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}  |
| Curl  \nabla \times \mathbf {A}  |  {\begin{matrix}\displaystyle \left({\partial A_{z} \over \partial y}-{\partial A_{y} \over \partial z}\right)\mathbf {\hat {x}} &+\\\displaystyle \left({\partial A_{x} \over \partial z}-{\partial A_{z} \over \partial x}\right)\mathbf {\hat {y}} &+\\\displaystyle \left({\partial A_{y} \over \partial x}-{\partial A_{x} \over \partial y}\right)\mathbf {\hat {z}} &\ \end{matrix}} |  {\begin{matrix}\displaystyle \left({1 \over \rho }{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}\right){\boldsymbol {\hat {\rho }}}&+\\\displaystyle \left({\partial A_{\rho } \over \partial z}-{\partial A_{z} \over \partial \rho }\right){\boldsymbol {\hat {\phi }}}&+\\\displaystyle {1 \over \rho }\left({\partial \left(\rho A_{\phi }\right) \over \partial \rho }-{\partial A_{\rho } \over \partial \phi }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}} |  {\begin{matrix}\displaystyle {1 \over r\sin \theta }\left({\partial  \over \partial \theta }\left(A_{\phi }\sin \theta \right)-{\partial A_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {r}}}&+\\\displaystyle {1 \over r}\left({1 \over \sin \theta }{\partial A_{r} \over \partial \phi }-{\partial  \over \partial r}\left(rA_{\phi }\right)\right){\boldsymbol {\hat {\theta }}}&+\\\displaystyle {1 \over r}\left({\partial  \over \partial r}\left(rA_{\theta }\right)-{\partial A_{r} \over \partial \theta }\right){\boldsymbol {\hat {\phi }}}&\ \end{matrix}} |  {\begin{matrix}\displaystyle \left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \tau }-{\partial A_{\tau } \over \partial z}\right){\boldsymbol {\hat {\sigma }}}&-\\\displaystyle \left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial A_{z} \over \partial \sigma }-{\partial A_{\sigma } \over \partial z}\right){\boldsymbol {\hat {\tau }}}&+\\\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right) \over \partial \tau }-{\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right) \over \partial \sigma }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}  |
| Laplace işlemcisi \Delta f=\nabla ^{2}f |  {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}} |  {1 \over \rho }{\partial  \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}} |  {1 \over r^{2}}{\partial  \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial  \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}} |  {\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}  |
| Vektör Laplasyeni  \Delta \mathbf {A} =\nabla ^{2}\mathbf {A}  |  \Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}}  |  {\begin{matrix}\displaystyle \left(\Delta A_{\rho }-{A_{\rho } \over \rho ^{2}}-{2 \over \rho ^{2}}{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {\rho }}}&+\\\displaystyle \left(\Delta A_{\phi }-{A_{\phi } \over \rho ^{2}}+{2 \over \rho ^{2}}{\partial A_{\rho } \over \partial \phi }\right){\boldsymbol {\hat {\phi }}}&+\\\displaystyle \left(\Delta A_{z}\right){\boldsymbol {\hat {z}}}&\ \end{matrix}} |  {\begin{matrix}\left(\Delta A_{r}-{2A_{r} \over r^{2}}-{2 \over r^{2}\sin \theta }{\partial \left(A_{\theta }\sin \theta \right) \over \partial \theta }-{2 \over r^{2}\sin \theta }{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {r}}}&+\\\left(\Delta A_{\theta }-{A_{\theta } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}}{\partial A_{r} \over \partial \theta }-{2\cos \theta  \over r^{2}\sin ^{2}\theta }{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {\theta }}}&+\\\left(\Delta A_{\phi }-{A_{\phi } \over r^{2}\sin ^{2}\theta }+{2 \over r^{2}\sin \theta }{\partial A_{r} \over \partial \phi }+{2\cos \theta  \over r^{2}\sin ^{2}\theta }{\partial A_{\theta } \over \partial \phi }\right){\boldsymbol {\hat {\phi }}}&\end{matrix}}  |  |
| Malzeme türevi  (\mathbf {A} \cdot \nabla )\mathbf {B}   |  {\begin{matrix}\displaystyle \left(A_{x}{\frac {\partial }{\partial x}}+A_{y}{\frac {\partial }{\partial y}}+A_{z}{\frac {\partial }{\partial z}}\right)B_{x}{\boldsymbol {\hat {x}}}+\\\displaystyle \left(A_{x}{\frac {\partial }{\partial x}}+A_{y}{\frac {\partial }{\partial y}}+A_{z}{\frac {\partial }{\partial z}}\right)B_{y}{\boldsymbol {\hat {y}}}+\\\displaystyle \left(A_{x}{\frac {\partial }{\partial x}}+A_{y}{\frac {\partial }{\partial y}}+A_{z}{\frac {\partial }{\partial z}}\right)B_{z}{\boldsymbol {\hat {z}}}\end{matrix}}  |  {\begin{matrix}\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \phi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\phi }B_{\phi }}{\rho }}\right){\boldsymbol {\hat {\rho }}}\!+\!\\\left(A_{\rho }{\frac {\partial B_{\phi }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\phi }}{\partial \phi }}+A_{z}{\frac {\partial B_{\phi }}{\partial z}}+{\frac {A_{\phi }B_{\rho }}{\rho }}\right){\boldsymbol {\hat {\phi }}}\!+\!\\\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{z}}{\partial \phi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right){\boldsymbol {\hat {z}}}\end{matrix}}  |  {\begin{matrix}\left(A_{r}{\frac {\partial B_{r}}{\partial r}}\!+\!{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}\!+\!{\frac {A_{\phi }}{r\sin(\theta )}}{\frac {\partial B_{r}}{\partial \phi }}\!-\!{\frac {A_{\theta }B_{\theta }\!+\!A_{\phi }B_{\phi }}{r}}\right){\boldsymbol {\hat {r}}}\!+\!\\\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}\!+\!{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}\!+\!{\frac {A_{\phi }}{r\sin(\theta )}}{\frac {\partial B_{\theta }}{\partial \phi }}\!+\!{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot(\theta )}{r}}\right){\boldsymbol {\hat {\theta }}}\!+\!\\\left(A_{r}{\frac {\partial B_{\phi }}{\partial r}}\!+\!{\frac {A_{\theta }}{r}}{\frac {\partial B_{\phi }}{\partial \theta }}\!+\!{\frac {A_{\phi }}{r\sin(\theta )}}{\frac {\partial B_{\phi }}{\partial \phi }}\!+\!{\frac {A_{\phi }B_{r}}{r}}\!+\!{\frac {A_{\phi }B_{\theta }\cot(\theta )}{r}}\right){\boldsymbol {\hat {\phi }}}\end{matrix}}  |  |
| Diferansiyel yer değiştirme |  {\begin{aligned}d\mathbf {l} &=d\mathbf {x} +d\mathbf {y} +d\mathbf {z} \\&=dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}} \end{aligned}} |  {\begin{aligned}d\mathbf {l} &=d{\boldsymbol {\rho }}+d{\boldsymbol {\phi }}+d\mathbf {z} \\&=d\rho {\boldsymbol {\hat {\rho }}}+\rho d\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}\end{aligned}} |  {\begin{aligned}d\mathbf {l} &=d\mathbf {r} +d{\boldsymbol {\theta }}+d{\boldsymbol {\phi }}\\&=dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}\end{aligned}} |  d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma {\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}d\tau {\boldsymbol {\hat {\tau }}}+dz{\boldsymbol {\hat {z}}}  |
| Diferansiyel yüzey normali |  {\begin{aligned}d\mathbf {S} &=d\mathbf {y} \times d\mathbf {z} +d\mathbf {z} \times d\mathbf {x} +d\mathbf {x} \times d\mathbf {y} \\&=dy\,dz\,\mathbf {\hat {x}} +dx\,dz\,\mathbf {\hat {y}} +dx\,dy\,\mathbf {\hat {z}} \end{aligned}} |  {\begin{aligned}d\mathbf {S} &=d{\boldsymbol {\phi }}\times d\mathbf {z} +d\mathbf {z} \times d{\boldsymbol {\rho }}+d{\boldsymbol {\rho }}\times d{\boldsymbol {\phi }}\\&=\rho \,d\phi \,dz\,{\boldsymbol {\hat {\rho }}}+d\rho \,dz\,{\boldsymbol {\hat {\phi }}}+\rho \,d\rho d\phi \,\mathbf {\hat {z}} \end{aligned}} |  {\begin{aligned}d\mathbf {S} &=d{\boldsymbol {\theta }}\times d{\boldsymbol {\phi }}+d{\boldsymbol {\phi }}\times d\mathbf {r} +d\mathbf {r} \times d{\boldsymbol {\theta }}\\&=r^{2}\sin \theta \,d\theta \,d\phi \,\mathbf {\hat {r}} +r\sin \theta \,d\phi \,dr\,{\boldsymbol {\hat {\theta }}}+r\,dr\,d\theta \,{\boldsymbol {\hat {\phi }}}\end{aligned}} |  {\begin{matrix}d\mathbf {S} =&{\sqrt {\sigma ^{2}+\tau ^{2}}},d\tau \,dz\,{\boldsymbol {\hat {\sigma }}}+\\&{\sqrt {\sigma ^{2}+\tau ^{2}}}d\sigma \,dz\,{\boldsymbol {\hat {\tau }}}+\\&\sigma ^{2}+\tau ^{2}d\sigma ,d\tau \,\mathbf {\hat {z}} \end{matrix}}  |
| Diferansiyel hacim |  dV=dx\,dy\,dz\, |  dV=\rho \,d\rho \,d\phi \,dz\, |  dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi \, |  dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,  |