Why did covariant nu change sides with mu in the inhomogenous maxwell equation?

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In summary, the inhomogenous Maxwell equations can be expressed as written in eq.(1.2), where the covariant "nu" changes sides with "mu" and becomes contravariant on the second term of the left hand side when taking the derivative of eq.(1.3) by \partial_{\nu}. This is due to the author commuting \partial^{\nu} and \partial_{\mu} in the second term of the left hand side.
  • #1
ercagpince
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[SOLVED] Inhomogenous maxwell equation

Homework Statement


In relativistic notation , the field strenght tensor can be expressed as (A is the vector potential) as on eq.(1.1) .

The inhomogenous Maxwell equations can be written as on eq.(1.2) .
Why did covariant "nu" changed sides with "mu" and become contravariant on the second term of left hand side of the equation 1.2 when one take the derivative of eq.(1.3) by
[tex]\partial_{\nu}[/tex]?


Homework Equations


[tex]F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}[/tex] (1.1)

[tex]\partial_{\mu}\partial^{\mu}A^{\nu}-\partial^{\nu}\partial_{\mu}A^{\mu}=\frac{4\Pi}{c}J^{\nu}[/tex] (1.2)

[tex]\partial_{\mu}F^{\mu\nu}=\frac{4\Pi}{c}J^{\nu}[/tex] (1.3)

The Attempt at a Solution


I tried to contract all terms in the eq.(1.2) , however , I couldn't find a quantitative way to solve the problem .
 
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  • #2
ercagpince said:
Why did covariant "nu" changed sides with "mu" and become contravariant on the second term of left hand side of the equation 1.2 when one take the derivative of eq.(1.3) by
[tex]\partial_{\nu}[/tex]?

The author didn't take the derivative [itex]\partial_{\nu}[/itex], he took the derivative [itex]\partial_{\mu}[/itex]. In the second term of the left hand side he simply commuted [itex]\partial^{\nu}[/itex] and [itex]\partial_{\mu}[/itex].
 
  • #3
thanks a lot!
 

1. What is the Inhomogenous Maxwell Equation?

The Inhomogenous Maxwell Equation is a set of four partial differential equations that describe the relationship between electric and magnetic fields and their sources, such as charges and currents. They are an essential part of classical electromagnetism and are used to understand and predict the behavior of electromagnetic waves.

2. How is the Inhomogenous Maxwell Equation different from the Homogeneous Maxwell Equation?

The Inhomogenous Maxwell Equation includes a term for the presence of electric charges and currents, while the Homogeneous Maxwell Equation does not. This term allows for the analysis of electromagnetic waves in the presence of sources, while the Homogeneous Maxwell Equation can only describe the behavior of waves in free space.

3. What are the four equations that make up the Inhomogenous Maxwell Equation?

The four equations are Gauss's Law, which relates electric fields to electric charges; Gauss's Law for Magnetism, which relates magnetic fields to magnetic charges; Faraday's Law, which describes the relationship between changing magnetic fields and induced electric fields; and Ampere's Law, which relates magnetic fields to electric currents.

4. What is the significance of the Inhomogenous Maxwell Equation?

The Inhomogenous Maxwell Equation is a fundamental part of electromagnetism and has many practical applications. It is used in the design and analysis of electronic circuits, communication systems, and electromagnetic devices. It also plays a crucial role in understanding the behavior of light and other electromagnetic waves in different materials and environments.

5. How is the Inhomogenous Maxwell Equation used in research and development?

The Inhomogenous Maxwell Equation is used extensively in research and development in various fields, including physics, engineering, and materials science. It is used to model and analyze electromagnetic phenomena, such as the behavior of light in optical fibers, the properties of materials, and the design of antennas and electronic devices. It also serves as a foundation for more advanced theories, such as quantum electrodynamics.

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