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louvig
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Help Understanding Equation 3.6 in Covariant Physics by Moataz H. Emam
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In summary, the conversation is about the transformation equation for a displacement vector in Moataz H. Emam's book, Covariant Physics. The author uses Einstein index notation to show the covariance of classical mechanics and the transformation of the position vector. The final equation, 3.6, has a typo and should read "j" instead of "i'". The derivation is straight-forward and involves using the transformation equations for primed coordinates. The author's point is to show that it transforms like a tensor and is therefore invariant.
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It's a bit difficult to read. Also, perhaps needs some context re the author's notation.
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louvig
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Sorry. I tried a screenshot from Kindle instead. I am able to click on it in my smartphone and make it full screen and is legible. The author is showing the covariance of classical mechanics using Einstein index notation. In this instance he is showing the transformation of the position vector which is straightforward and then the transformation of the derivative of the position vector. His point is to show ot transforms like a tensor and is therefore invariant.PeroK said:
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malawi_glenn
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eq 3.6 has a typo, this index should read ##j## https://web.cortland.edu/moataz.emam/
The derivation is straight-forward:
Use that ##\hat{ \textbf{g}}_{i'} = \lambda^k_{i'} \hat{ \textbf{e}}_k ## and ##x^{i'} = \lambda^{i'}_j x^j##.
We get ## d\hat{ \textbf{g}}_{i'} = \hat{ \textbf{e}}_k d \lambda^k_{i'} ## and ##x^{i'} = x^j d\lambda^{i'}_j + \lambda^{i'}_j dx^j##.
And you will obtain the final step in that equation.
The derivation is straight-forward:
Use that ##\hat{ \textbf{g}}_{i'} = \lambda^k_{i'} \hat{ \textbf{e}}_k ## and ##x^{i'} = \lambda^{i'}_j x^j##.
We get ## d\hat{ \textbf{g}}_{i'} = \hat{ \textbf{e}}_k d \lambda^k_{i'} ## and ##x^{i'} = x^j d\lambda^{i'}_j + \lambda^{i'}_j dx^j##.
And you will obtain the final step in that equation.
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louvig
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Thank you so much. Makes sense.
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Moataz Emam
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louvig said:
Everything with primed coordinates was replaced with its transformation. So x’=lambda x and so on.
FAQ: Help Understanding Equation 3.6 in Covariant Physics by Moataz H. Emam
What is the primary focus of Equation 3.6 in Covariant Physics by Moataz H. Emam?
Equation 3.6 in Covariant Physics by Moataz H. Emam primarily deals with the transformation properties of tensor fields under coordinate transformations. It is a fundamental concept in understanding how physical laws remain invariant in different reference frames.
How does Equation 3.6 relate to the concept of covariance in physics?
Equation 3.6 exemplifies the principle of covariance, which asserts that the form of physical laws should remain unchanged under a change of coordinates. This equation mathematically demonstrates how tensor quantities transform, ensuring that the laws of physics are the same in all coordinate systems.
Can you provide a step-by-step derivation of Equation 3.6?
The derivation of Equation 3.6 typically involves starting from the definition of a tensor and applying the rules of coordinate transformations. One would need to use the chain rule for partial derivatives and the properties of the metric tensor to show how each component of the tensor transforms under a change of coordinates.
What are the common mistakes to avoid when working with Equation 3.6?
Common mistakes include incorrect application of the chain rule, neglecting the metric tensor's role in raising and lowering indices, and misinterpreting the transformation properties of different types of tensors (contravariant, covariant, and mixed). Ensuring proper index notation and careful algebraic manipulation are crucial.
How is Equation 3.6 applied in practical problems in physics?
Equation 3.6 is applied in various areas of physics, including general relativity, fluid dynamics, and electromagnetism. It helps in transforming equations between different coordinate systems, making it easier to solve complex physical problems by choosing the most convenient frame of reference.