Usuário(a):Giro720/testes2

	Coordenadas cartesianas (x, y, z)	Coordenadas cilíndricas (ρ, ϕ, z)	Coordenadas esféricas (r, θ, ϕ)
Definição das coordenadas	${\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}$	${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}$	${\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}$
	${\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}$	${\begin{aligned}\rho &=r\sin \theta \\\phi &=\phi \\z&=r\cos \theta \end{aligned}}$	${\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}$
Definição do vetor unitário	${\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\mathbf {\hat {r}} &={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z\mathbf {\hat {z}} }{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right)\mathbf {\hat {z}} }{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$
	${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}$	${\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho {\boldsymbol {\hat {\rho }}}+z\mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {z{\boldsymbol {\hat {\rho }}}-\rho \mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}}$
Campo vetorial $\mathbf {A}$	$A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}\mathbf {\hat {z}}$	$A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}\mathbf {\hat {z}}$	$A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}$
Gradiente de um campo escalar $f$: $\nabla f$	${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}\mathbf {\hat {z}}$	${\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}$
Divergência: $\nabla \cdot \mathbf {A}$	${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$	${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$	${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$
Rotacional: $\nabla \times \mathbf {A}$	${\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}+\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}+\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right)&{\boldsymbol {\hat {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\boldsymbol {\hat {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right)&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\boldsymbol {\hat {r}}}\\+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right)&{\boldsymbol {\hat {\theta }}}\\+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\boldsymbol {\hat {\phi }}}\end{aligned}}$
Laplaciano: $\nabla ^{2}f$	${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$	${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$
Deslocamento diferencial	$d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,\mathbf {\hat {z}}$	$d\mathbf {l} =d\rho \,{\boldsymbol {\hat {\rho }}}+\rho \,d\phi \,{\boldsymbol {\hat {\phi }}}+dz\,\mathbf {\hat {z}}$	$d\mathbf {l} =dr\,\mathbf {\hat {r}} +r\,d\theta \,{\boldsymbol {\hat {\theta }}}+r\,\sin \theta \,d\phi \,{\boldsymbol {\hat {\phi }}}$
Diferencial de àrea orientada $d\mathbf {S}$	${\begin{aligned}dy\,dz&{\hat {\mathbf {x} }}\\+dx\,dz&{\hat {\mathbf {y} }}\\+dx\,dy&\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}\rho \,d\phi \,dz&{\boldsymbol {\hat {\rho }}}\\+d\rho \,dz&{\boldsymbol {\hat {\phi }}}\\+\rho \,d\rho \,d\phi &\mathbf {\hat {z}} \end{aligned}}$	${\begin{aligned}r^{2}\sin \theta \,d\theta \,d\phi &\mathbf {\hat {r}} \\+r\sin \theta \,dr\,d\phi &{\boldsymbol {\hat {\theta }}}\\+r\,dr\,d\theta &{\boldsymbol {\hat {\phi }}}\end{aligned}}$
Diferencial de volume $dV$	$dx\,dy\,dz$	$\rho \,d\rho \,d\phi \,dz$	$r^{2}\sin \theta \,dr\,d\theta \,d\phi$