işlem
Kartezyen koordinatlar (x, y,z)
Silindirik koordinatlar (ρ,φ,z)
Küresel koordinatlar (r,θ,φ)
Parabolik silindrik koordinatlar (σ,τ,z)
Koordinat Tanımları
ρ
=
x
2
+
y
2
ϕ
=
arctan
(
y
/
x
)
z
=
z
{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}}
x
=
ρ
cos
ϕ
y
=
ρ
sin
ϕ
z
=
z
{\displaystyle {\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}}
x
=
r
sin
θ
cos
ϕ
y
=
r
sin
θ
sin
ϕ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}}
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}}
r
=
x
2
+
y
2
+
z
2
θ
=
arccos
(
z
/
r
)
ϕ
=
arctan
(
y
/
x
)
{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}}
r
=
ρ
2
+
z
2
θ
=
arctan
(
ρ
/
z
)
ϕ
=
ϕ
{\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}}
ρ
=
r
sin
(
θ
)
ϕ
=
ϕ
z
=
r
cos
(
θ
)
{\displaystyle {\begin{aligned}\rho &=r\sin(\theta )\\\phi &=\phi \\z&=r\cos(\theta )\end{aligned}}}
ρ
cos
ϕ
=
σ
τ
ρ
sin
ϕ
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\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
ρ
^
=
x
x
^
+
y
y
^
x
2
+
y
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\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}}}
x
^
=
cos
ϕ
ρ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
ϕ
ρ
^
+
cos
ϕ
ϕ
^
z
^
=
z
^
{\displaystyle {\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}}}
x
^
=
cos
ϕ
(
sin
θ
r
^
+
cos
θ
θ
^
)
−
sin
ϕ
ϕ
^
y
^
=
sin
ϕ
(
sin
θ
r
^
+
c
o
s
θ
θ
^
)
+
cos
ϕ
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\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}}}
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{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}}}
r
^
=
x
x
^
+
y
y
^
+
z
z
^
x
2
+
y
2
+
z
2
θ
^
=
z
(
x
x
^
+
y
y
^
)
−
(
x
2
+
y
2
)
z
^
x
2
+
y
2
+
z
2
x
2
+
y
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
{\displaystyle {\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}}}
r
^
=
ρ
ρ
2
+
z
2
ρ
^
+
z
ρ
2
+
z
2
z
^
θ
^
=
z
ρ
2
+
z
2
ρ
^
−
ρ
ρ
2
+
z
2
z
^
ϕ
^
=
ϕ
^
{\displaystyle {\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}}}
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
ϕ
^
=
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\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}}}
{\displaystyle {\begin{matrix}\end{matrix}}}
Bir vektör alanı
A
{\displaystyle \mathbf {A} }
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} }
A
ρ
ρ
^
+
A
ϕ
ϕ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}{\boldsymbol {\hat {z}}}}
A
r
r
^
+
A
θ
θ
^
+
A
ϕ
ϕ
^
{\displaystyle A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}}
A
σ
σ
^
+
A
τ
τ
^
+
A
ϕ
z
^
{\displaystyle A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }{\boldsymbol {\hat {z}}}}
Gradyan
∇
f
{\displaystyle \nabla f}
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}\mathbf {\hat {x}} +{\partial f \over \partial y}\mathbf {\hat {y}} +{\partial f \over \partial z}\mathbf {\hat {z}} }
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
ϕ
ϕ
^
+
∂
f
∂
z
z
^
{\displaystyle {\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}}}}
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
ϕ
ϕ
^
{\displaystyle {\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 }}}}
1
σ
2
+
τ
2
∂
f
∂
σ
σ
^
+
1
σ
2
+
τ
2
∂
f
∂
τ
τ
^
+
∂
f
∂
z
z
^
{\displaystyle {\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
∇
⋅
A
{\displaystyle \nabla \cdot \mathbf {A} }
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\displaystyle {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
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
ϕ
∂
ϕ
{\displaystyle {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 }}
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
σ
+
∂
(
σ
2
+
τ
2
A
τ
)
∂
τ
)
+
∂
A
z
∂
z
{\displaystyle {\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
∇
×
A
{\displaystyle \nabla \times \mathbf {A} }
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle {\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}}}
(
1
ρ
∂
A
z
∂
ϕ
−
∂
A
ϕ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
ϕ
^
+
1
ρ
(
∂
(
ρ
A
ϕ
)
∂
ρ
−
∂
A
ρ
∂
ϕ
)
z
^
{\displaystyle {\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}}}
1
r
sin
θ
(
∂
∂
θ
(
A
ϕ
sin
θ
)
−
∂
A
θ
∂
ϕ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
ϕ
−
∂
∂
r
(
r
A
ϕ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
ϕ
^
{\displaystyle {\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}}}
(
1
σ
2
+
τ
2
∂
A
z
∂
τ
−
∂
A
τ
∂
z
)
σ
^
−
(
1
σ
2
+
τ
2
∂
A
z
∂
σ
−
∂
A
σ
∂
z
)
τ
^
+
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
τ
−
∂
(
σ
2
+
τ
2
A
τ
)
∂
σ
)
z
^
{\displaystyle {\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
Δ
f
=
∇
2
f
{\displaystyle \Delta f=\nabla ^{2}f}
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
ϕ
2
+
∂
2
f
∂
z
2
{\displaystyle {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
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
ϕ
2
{\displaystyle {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}}}
1
σ
2
+
τ
2
(
∂
2
f
∂
σ
2
+
∂
2
f
∂
τ
2
)
+
∂
2
f
∂
z
2
{\displaystyle {\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
Δ
A
=
∇
2
A
{\displaystyle \Delta \mathbf {A} =\nabla ^{2}\mathbf {A} }
Δ
A
x
x
^
+
Δ
A
y
y
^
+
Δ
A
z
z
^
{\displaystyle \Delta A_{x}\mathbf {\hat {x}} +\Delta A_{y}\mathbf {\hat {y}} +\Delta A_{z}\mathbf {\hat {z}} }
(
Δ
A
ρ
−
A
ρ
ρ
2
−
2
ρ
2
∂
A
ϕ
∂
ϕ
)
ρ
^
+
(
Δ
A
ϕ
−
A
ϕ
ρ
2
+
2
ρ
2
∂
A
ρ
∂
ϕ
)
ϕ
^
+
(
Δ
A
z
)
z
^
{\displaystyle {\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}}}
(
Δ
A
r
−
2
A
r
r
2
−
2
r
2
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
−
2
r
2
sin
θ
∂
A
ϕ
∂
ϕ
)
r
^
+
(
Δ
A
θ
−
A
θ
r
2
sin
2
θ
+
2
r
2
∂
A
r
∂
θ
−
2
cos
θ
r
2
sin
2
θ
∂
A
ϕ
∂
ϕ
)
θ
^
+
(
Δ
A
ϕ
−
A
ϕ
r
2
sin
2
θ
+
2
r
2
sin
θ
∂
A
r
∂
ϕ
+
2
cos
θ
r
2
sin
2
θ
∂
A
θ
∂
ϕ
)
ϕ
^
{\displaystyle {\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 [ 1]
(
A
⋅
∇
)
B
{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} }
(
A
x
∂
∂
x
+
A
y
∂
∂
y
+
A
z
∂
∂
z
)
B
x
x
^
+
(
A
x
∂
∂
x
+
A
y
∂
∂
y
+
A
z
∂
∂
z
)
B
y
y
^
+
(
A
x
∂
∂
x
+
A
y
∂
∂
y
+
A
z
∂
∂
z
)
B
z
z
^
{\displaystyle {\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}}}
(
A
ρ
∂
B
ρ
∂
ρ
+
A
ϕ
ρ
∂
B
ρ
∂
ϕ
+
A
z
∂
B
ρ
∂
z
−
A
ϕ
B
ϕ
ρ
)
ρ
^
+
(
A
ρ
∂
B
ϕ
∂
ρ
+
A
ϕ
ρ
∂
B
ϕ
∂
ϕ
+
A
z
∂
B
ϕ
∂
z
+
A
ϕ
B
ρ
ρ
)
ϕ
^
+
(
A
ρ
∂
B
z
∂
ρ
+
A
ϕ
ρ
∂
B
z
∂
ϕ
+
A
z
∂
B
z
∂
z
)
z
^
{\displaystyle {\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}}}
(
A
r
∂
B
r
∂
r
+
A
θ
r
∂
B
r
∂
θ
+
A
ϕ
r
sin
(
θ
)
∂
B
r
∂
ϕ
−
A
θ
B
θ
+
A
ϕ
B
ϕ
r
)
r
^
+
(
A
r
∂
B
θ
∂
r
+
A
θ
r
∂
B
θ
∂
θ
+
A
ϕ
r
sin
(
θ
)
∂
B
θ
∂
ϕ
+
A
θ
B
r
r
−
A
ϕ
B
ϕ
cot
(
θ
)
r
)
θ
^
+
(
A
r
∂
B
ϕ
∂
r
+
A
θ
r
∂
B
ϕ
∂
θ
+
A
ϕ
r
sin
(
θ
)
∂
B
ϕ
∂
ϕ
+
A
ϕ
B
r
r
+
A
ϕ
B
θ
cot
(
θ
)
r
)
ϕ
^
{\displaystyle {\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
d
l
=
d
x
+
d
y
+
d
z
=
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle {\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}}}
d
l
=
d
ρ
+
d
ϕ
+
d
z
=
d
ρ
ρ
^
+
ρ
d
ϕ
ϕ
^
+
d
z
z
^
{\displaystyle {\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}}}
d
l
=
d
r
+
d
θ
+
d
ϕ
=
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
ϕ
ϕ
^
{\displaystyle {\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
l
=
σ
2
+
τ
2
d
σ
σ
^
+
σ
2
+
τ
2
d
τ
τ
^
+
d
z
z
^
{\displaystyle 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
d
S
=
d
y
×
d
z
+
d
z
×
d
x
+
d
x
×
d
y
=
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle {\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}}}
d
S
=
d
ϕ
×
d
z
+
d
z
×
d
ρ
+
d
ρ
×
d
ϕ
=
ρ
d
ϕ
d
z
ρ
^
+
d
ρ
d
z
ϕ
^
+
ρ
d
ρ
d
ϕ
z
^
{\displaystyle {\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}}}
d
S
=
d
θ
×
d
ϕ
+
d
ϕ
×
d
r
+
d
r
×
d
θ
=
r
2
sin
θ
d
θ
d
ϕ
r
^
+
r
sin
θ
d
ϕ
d
r
θ
^
+
r
d
r
d
θ
ϕ
^
{\displaystyle {\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}}}
d
S
=
σ
2
+
τ
2
,
d
τ
d
z
σ
^
+
σ
2
+
τ
2
d
σ
d
z
τ
^
+
σ
2
+
τ
2
d
σ
,
d
τ
z
^
{\displaystyle {\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
d
V
=
d
x
d
y
d
z
{\displaystyle dV=dx\,dy\,dz\,}
d
V
=
ρ
d
ρ
d
ϕ
d
z
{\displaystyle dV=\rho \,d\rho \,d\phi \,dz\,}
d
V
=
r
2
sin
θ
d
r
d
θ
d
ϕ
{\displaystyle dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi \,}
d
V
=
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
,
{\displaystyle dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,}
önemli birtakım hesaplama kuralları:
div
grad
f
=
∇
⋅
(
∇
f
)
=
∇
2
f
=
Δ
f
{\displaystyle \operatorname {div} \ \operatorname {grad} f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}
Laplasyen )
curl
grad
f
=
∇
×
(
∇
f
)
=
0
{\displaystyle \operatorname {curl} \ \operatorname {grad} f=\nabla \times (\nabla f)=\mathbf {0} }
div
curl
A
=
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div} \ \operatorname {curl} \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}
curl
curl
A
=
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {curl} \ \operatorname {curl} \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
Δ
(
f
g
)
=
f
Δ
g
+
2
∇
f
⋅
∇
g
+
g
Δ
f
{\displaystyle \Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f}
