Silindirik ve küresel koordinatlarda del

Bu liste eğrisel koordinat sistemleri ile çalışılırken genel olarak kullanılan vektör hesabı formüllerinin bir listesidir.

Not

• Bu sayfada standart fizik gösterim kullanır. küresel koordinatlar için, θ açısı yarıçap vektörünün z ekseni ile olan açısıdır ve Söz konusu noktaya orijinden bağlanır. ϕ açısı yarıçap vektörünün x-y yüzeyine izdüşümü ile ve x ekseni ile olan açıdır. Bazı kaynaklar θ ve ϕ yi ters tanıtırlar, bu anlam bağlamında böyle bir bağlantı kurulmamalıdır.
• atan2(y, x) fonsiyonu kendi etki ve görüntü nedeniyle matematiksel fonksiyon arctan(y/x) yerine kullanılabilir, klasik arctan(y/x) görüntüsü (-π/2, +π/2)aralığında idi, buradaki atan2(y, x) (-π, π] aralığındadır. (Küresel koordinatlarda Del için ifadelerin düzeltilmesi gerekebilir)
• Dönüşümler kartezyen koordinatlardan silindirik ve küreseledir.

del operatörü ile Silindirik küresel ve parabolik silindirik koordinatlar tablosu

işlem	Kartezyen koordinatlar (x, y,z)	Silindirik koordinatlar (ρ,φ,z)	Küresel koordinatlar (r,θ,φ)	Parabolik silindrik koordinatlar (σ,τ,z)
Koordinat Tanımları	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}$	${\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}$	${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$	${\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$
	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}$	${\begin{aligned}\rho &=r\sin(\theta )\\\phi &=\phi \\z&=r\cos(\theta )\end{aligned}}$	${\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}$
Birim Vektölerin Tanımları	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x\mathbf {\hat {x}} +y\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\mathbf {\hat {x}} &=\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\mathbf {\hat {x}} &=\cos \phi \left(\sin \theta {\boldsymbol {\hat {r}}}+\cos \theta {\boldsymbol {\hat {\theta }}}\right)-\sin \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {y}} &=\sin \phi \left(\sin \theta {\boldsymbol {\hat {r}}}+\ cos\theta {\boldsymbol {\hat {\theta }}}\right)+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\sigma }}}&={\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} -{\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\{\boldsymbol {\hat {\tau }}}&={\frac {\sigma }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {x}} +{\frac {\tau }{\sqrt {\tau ^{2}+\sigma ^{2}}}}\mathbf {\hat {y}} \\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$
	${\begin{aligned}\mathbf {\hat {r}} &={\frac {x\mathbf {\hat {x}} \!+\!y\mathbf {\hat {y}} \!+\!z\mathbf {\hat {z}} }{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {z\left(x\mathbf {\hat {x}} \!+\!y\mathbf {\hat {y}} \right)\!-\!\left(x^{2}+y^{2}\right)\mathbf {\hat {z}} }{{\sqrt {x^{2}+y^{2}+z^{2}}}{\sqrt {x^{2}+y^{2}}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y\mathbf {\hat {x}} +x\mathbf {\hat {y}} }{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$	${\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho }{\sqrt {\rho ^{2}+z^{2}}}}{\boldsymbol {\hat {\rho }}}+{\frac {z}{\sqrt {\rho ^{2}+z^{2}}}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\theta }}}&={\frac {z}{\sqrt {\rho ^{2}+z^{2}}}}{\boldsymbol {\hat {\rho }}}-{\frac {\rho }{\sqrt {\rho ^{2}+z^{2}}}}\mathbf {\hat {z}} \\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{matrix}\end{matrix}}$
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	$dV=dx\,dy\,dz\,$	$dV=\rho \,d\rho \,d\phi \,dz\,$	$dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi \,$	$dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,$
önemli birtakım hesaplama kuralları: $\operatorname {div} \ \operatorname {grad} f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f$ (Laplasyen) $\operatorname {curl} \ \operatorname {grad} f=\nabla \times (\nabla f)=\mathbf {0}$ $\operatorname {div} \ \operatorname {curl} \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0$ $\operatorname {curl} \ \operatorname {curl} \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ (Vektör çarpımı için Lagrange formülünü kullanarak) $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$

Ayrıca bakınız

• Del işlemcisi
• Ortogonal koordinatlar
• Eğrisel koordinatlar
• Silindirik ve küresel koordinatlarda vektör alanı
• Birim vektör

Dış bağlantılar

• Maxima Computer Algebra system scripts6 Eylül 2014 tarihinde Wayback Machine sitesinde arşivlendi. to generate some of these operators in cylindrical and spherical coordinates.