संक्रिया (Operation)
कार्तीय निर्देशांक (x,y,z)
बेलनाकार निर्देशांक (s,φ,z)
गोलीय निर्देशांक (r,θ,φ)
परबलीय बेलनाकार निर्देशांक (ο,τ,z)
निर्देशांकों
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}}}
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}}}
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{matrix}x&=&\sigma \tau \\y&=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
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}}}
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}}}
s
cos
ϕ
=
σ
τ
s
sin
ϕ
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{matrix}s\cos \phi &=&\sigma \tau \\s\sin \phi &=&{\frac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=&z\end{matrix}}}
सदिशों
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}}}
A vector field
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
s
s
^
+
A
ϕ
ϕ
^
+
A
z
z
^
{\displaystyle A_{s}{\boldsymbol {\hat {s}}}+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}}}}
Gradient
∇
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
∂
s
s
^
+
1
s
∂
f
∂
ϕ
ϕ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial s}{\boldsymbol {\hat {s}}}+{1 \over s}{\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}}}}
Divergence
∇
⋅
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
s
∂
(
s
A
s
)
∂
s
+
1
s
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\displaystyle {1 \over s}{\partial \left(sA_{s}\right) \over \partial s}+{1 \over s}{\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
∂
A
σ
∂
σ
+
1
σ
2
+
τ
2
∂
A
τ
∂
τ
+
∂
A
z
∂
z
{\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2}}}{\partial A_{\sigma } \over \partial \sigma }+{\frac {1}{\sigma ^{2}+\tau ^{2}}}{\partial A_{\tau } \over \partial \tau }+{\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
s
∂
A
z
∂
ϕ
−
∂
A
ϕ
∂
z
)
s
^
+
(
∂
A
s
∂
z
−
∂
A
z
∂
s
)
ϕ
^
+
1
s
(
∂
(
s
A
ϕ
)
∂
s
−
∂
A
s
∂
ϕ
)
z
^
{\displaystyle {\begin{matrix}\displaystyle \left({1 \over s}{\partial A_{z} \over \partial \phi }-{\partial A_{\phi } \over \partial z}\right){\boldsymbol {\hat {s}}}&+\\\displaystyle \left({\partial A_{s} \over \partial z}-{\partial A_{z} \over \partial s}\right){\boldsymbol {\hat {\phi }}}&+\\\displaystyle {1 \over s}\left({\partial \left(sA_{\phi }\right) \over \partial s}-{\partial A_{s} \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
(
∂
(
s
A
ϕ
)
∂
s
−
∂
A
s
∂
ϕ
)
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(sA_{\phi }\right) \over \partial s}-{\partial A_{s} \over \partial \phi }\right){\boldsymbol {\hat {z}}}&\ \end{matrix}}}
Laplace operator
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
s
∂
∂
s
(
s
∂
f
∂
s
)
+
1
s
2
∂
2
f
∂
ϕ
2
+
∂
2
f
∂
z
2
{\displaystyle {1 \over s}{\partial \over \partial s}\left(s{\partial f \over \partial s}\right)+{1 \over s^{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}}}}
Vector Laplacian
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
s
−
A
s
s
2
−
2
s
2
∂
A
ϕ
∂
ϕ
)
s
^
+
(
Δ
A
ϕ
−
A
ϕ
s
2
+
2
s
2
∂
A
s
∂
ϕ
)
ϕ
^
+
(
Δ
A
z
)
z
^
{\displaystyle {\begin{matrix}\displaystyle \left(\Delta A_{s}-{A_{s} \over s^{2}}-{2 \over s^{2}}{\partial A_{\phi } \over \partial \phi }\right){\boldsymbol {\hat {s}}}&+\\\displaystyle \left(\Delta A_{\phi }-{A_{\phi } \over s^{2}}+{2 \over s^{2}}{\partial A_{s} \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}}}
Differential displacement
d
l
=
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =dx\mathbf {\hat {x}} +dy\mathbf {\hat {y}} +dz\mathbf {\hat {z}} }
d
l
=
d
s
s
^
+
s
d
ϕ
ϕ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =ds{\boldsymbol {\hat {s}}}+sd\phi {\boldsymbol {\hat {\phi }}}+dz{\boldsymbol {\hat {z}}}}
d
l
=
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
ϕ
ϕ
^
{\displaystyle d\mathbf {l} =dr\mathbf {\hat {r}} +rd\theta {\boldsymbol {\hat {\theta }}}+r\sin \theta d\phi {\boldsymbol {\hat {\phi }}}}
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}}}}
Differential normal area
d
S
=
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&dy\,dz\,\mathbf {\hat {x}} +\\&dx\,dz\,\mathbf {\hat {y}} +\\&dx\,dy\,\mathbf {\hat {z}} \end{matrix}}}
d
S
=
s
d
ϕ
d
z
s
^
+
d
s
d
z
ϕ
^
+
s
d
s
d
ϕ
z
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&s\,d\phi \,dz\,{\boldsymbol {\hat {s}}}+\\&ds\,dz\,{\boldsymbol {\hat {\phi }}}+\\&s\,dsd\phi \,\mathbf {\hat {z}} \end{matrix}}}
d
S
=
r
2
sin
θ
d
θ
d
ϕ
r
^
+
r
sin
θ
d
r
d
ϕ
θ
^
+
r
d
r
d
θ
ϕ
^
{\displaystyle {\begin{matrix}d\mathbf {S} =&r^{2}\sin \theta \,d\theta \,d\phi \,\mathbf {\hat {r}} +\\&r\sin \theta \,dr\,d\phi \,{\boldsymbol {\hat {\theta }}}+\\&r\,dr\,d\theta \,{\boldsymbol {\hat {\phi }}}\end{matrix}}}
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}}}
Differential volume
d
τ
=
d
x
d
y
d
z
{\displaystyle d\tau =dx\,dy\,dz\,}
d
τ
=
s
d
s
d
ϕ
d
z
{\displaystyle d\tau =s\,ds\,d\phi \,dz\,}
d
τ
=
r
2
sin
θ
d
r
d
θ
d
ϕ
{\displaystyle d\tau =r^{2}\sin \theta \,dr\,d\theta \,d\phi \,}
d
τ
=
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
,
{\displaystyle d\tau =\left(\sigma ^{2}+\tau ^{2}\right)d\sigma d\tau dz,}
डेल संक्रिया के कुछ असरल नियम:
d
i
v
g
r
a
d
f
=
∇
⋅
(
∇
f
)
=
∇
2
f
=
Δ
f
{\displaystyle \operatorname {div\ grad\ } f=\nabla \cdot (\nabla f)=\nabla ^{2}f=\Delta f}
Laplacian )
c
u
r
l
g
r
a
d
f
=
∇
×
(
∇
f
)
=
0
{\displaystyle \operatorname {curl\ grad\ } f=\nabla \times (\nabla f)=0}
d
i
v
c
u
r
l
A
=
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div\ curl\ } \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}
c
u
r
l
c
u
r
l
A
=
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {curl\ curl\ } \mathbf {A} =\nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
cross product )
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}
