Help with Derivation: u^-1(\partial_\mu u)u^-1 = \partial_\mu(u^-1)

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In summary, the derivative of u^-1(\partial_\mu u) can be found using the chain rule, where u^-1 is the inverse of u. This expression is a useful tool for solving differential equations involving u, as it allows us to simplify the equation and solve for u^-1, which can then be used to find the solution for u. This derivation can be applied to any function that has an inverse, and is a direct application of the inverse function theorem. It is commonly used in physics, engineering, and optimization problems.
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Kontilera
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Hello there!
I'm trying to read a set of notes on the standard model and the author states the following equality without any explanation:
[tex]-u^{-1}(\partial_\mu u)u^{-1} = \partial_\mu(u^{-1}), [/tex]
u is here a NxN representation of some continuous gauge group and is a function over spacetime.
Does this equality hold and in that case why?

Thanks!
// Kontilera
 
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Solved it!
Thanks!
 

FAQ: Help with Derivation: u^-1(\partial_\mu u)u^-1 = \partial_\mu(u^-1)

How do you derive u^-1(\partial_\mu u)?

The derivative of u^-1(\partial_\mu u) can be found using the chain rule. First, rewrite the expression as u^-1(u^\mu), where u^\mu is the inverse of u. Then, apply the chain rule to get u^-1(\partial_\mu u) = \partial_\mu(u^-1).

What is the purpose of u^-1(\partial_\mu u)u^-1 = \partial_\mu(u^-1)?

This expression is a useful tool for solving differential equations involving u. It allows us to simplify the equation and solve for u^-1, which can then be used to find the solution for u.

Can this derivation be applied to any function, or only to u?

This derivation can be applied to any function that has an inverse, not just u. However, the specific steps may vary depending on the function.

How does this derivation relate to the inverse function theorem?

The inverse function theorem states that if a function has a continuous inverse, then its derivative at a point is equal to the inverse of the derivative of its inverse at the corresponding point. This derivation is a direct application of this theorem.

Are there any other applications of this derivation in science?

Yes, this derivation is commonly used in physics and engineering to solve differential equations involving inverse functions. It is also used in optimization problems, where the inverse function is used to find the optimal solution.

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