Orthogonality from infinitesimal small rotation

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SUMMARY

The discussion centers on the mathematical concept of orthogonality in relation to infinitesimal rotations, specifically addressing the invariance of vector lengths under such transformations. The second equality is derived from the first using the chain rule, with the assumption that the variation of the metric tensor, denoted as gik, is zero. Participants clarify that gik represents the inner product of the basis vectors, <ei,ek>, and explore the implications of time dependence on the vector vik.

PREREQUISITES
  • Understanding of differential geometry concepts, particularly metric tensors.
  • Familiarity with the chain rule in calculus.
  • Knowledge of vector spaces and inner products.
  • Basic principles of rotation matrices in linear algebra.
NEXT STEPS
  • Study the properties of metric tensors in differential geometry.
  • Learn about the application of the chain rule in higher dimensions.
  • Explore the concept of infinitesimal transformations in physics.
  • Investigate the role of inner products in vector spaces and their geometric interpretations.
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Mathematicians, physicists, and students studying differential geometry or linear algebra, particularly those interested in the implications of rotations and orthogonality in vector spaces.

Warlord_
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Hello buddies,

Could someone please help me to understand where the second and the third equalities came from?
Thanks,

Page-question-tensor.jpg
 
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Um... Ok, it's a little difficult to know what level of detail you want.

The first equation essentially says that the length of a vector does not change under rotations. The second one follows from the first one using the chain rule, and assuming delta of gik is zero.
 
Thanks for answering,

gik is basically the inner product of the base set vector, i.e., <ei,ek>.
- Could you please explicit the chain rule?
- is vik time dependent? And since it is an infinitely small matrix, the ##\delta v^k_i = I##?

Thanks
 

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