Difference between revisions of "4-gradient"

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<center><math>\nabla_\mu=\frac{\partial}{\partial x_\mu}=\partial_\mu</math></center>
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From the use of the Minkowski metric, converting between contravariant and covariant
  
  
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<center><math>\mathbf x_{\mu} \equiv \eta_{\mu}^{\mu} \mathbf x^{\mu}</math></center>
  
<center><math>\mathbf \partial_\mu \equiv \Biggl [ \frac{\partial}{\partial t}\quad \frac{\partial}{\partial x}\quad \frac{\partial}{\partial y}\quad \frac{\partial}{\partial z}\Biggr ]=\Biggl [\frac{\partial}{\partial x_0}\quad \frac{\partial}{\partial x_1}\quad \frac{\partial}{\partial x_2}\quad \frac{\partial}{\partial x_3}\Biggr ]</math></center>
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<center><math>\nabla_{\mu}=\partial_{\mu}</math></center>
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<center><math>\mathbf \partial_\mu \equiv \Biggl [ \frac{\partial}{\partial t}\quad \frac{\partial}{\partial x}\quad \frac{\partial}{\partial y}\quad \frac{\partial}{\partial z}\Biggr ]=\Biggl [\frac{\partial}{\partial x^0}\quad \frac{\partial}{\partial x^1}\quad \frac{\partial}{\partial x^2}\quad \frac{\partial}{\partial x^3}\Biggr ]</math></center>

Revision as of 23:57, 9 July 2017

From the use of the Minkowski metric, converting between contravariant and covariant


[math]\mathbf x_{\mu} \equiv \eta_{\mu}^{\mu} \mathbf x^{\mu}[/math]


[math]\nabla_{\mu}=\partial_{\mu}[/math]


[math]\mathbf \partial_\mu \equiv \Biggl [ \frac{\partial}{\partial t}\quad \frac{\partial}{\partial x}\quad \frac{\partial}{\partial y}\quad \frac{\partial}{\partial z}\Biggr ]=\Biggl [\frac{\partial}{\partial x^0}\quad \frac{\partial}{\partial x^1}\quad \frac{\partial}{\partial x^2}\quad \frac{\partial}{\partial x^3}\Biggr ][/math]