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In summary, there are several sources that discuss the representation of Christoffel symbols as a linear combination of products of the metric tensor and its derivative. However, the derivation of this representation is often done in a confusing manner. The covariant derivative of the metric tensor is equal to zero, but this does not make it a tensor. To make it a tensor, the Christoffel symbols must be subtracted from the covariant derivative. There are also sources that provide a more physically intuitive proof, such as chapter 3 of Wald's GR book and "Riemannian manifolds: an introduction to curvature" by John Lee. Additionally, the proof can be found in MTW exercise 8.15.

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Check out chapter 3 of Wald's GR book.

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zwoodrow said:I am trying to make a more physically intuitive proof based off the covariant derivative of the metric tensor being equal to zero. Has anyone seen this proof somewhere i haven't got it to work out and i am looking go help.

Yes, you can find it in MTW exercise 8.15. It has an outline solution too.

Christoffel symbols are mathematical quantities used in differential geometry to describe the curvature of a space. They are used in the covariant derivative of a metric tensor, which is a way to measure how a vector changes as it moves along a curve.

Christoffel symbols are important because they allow us to describe the curvature of a space and calculate how vectors change as they move along a curve. They are also used in the equations of general relativity, which describe the behavior of gravity.

The intuitive proof for the covariant derivative of a metric tensor using Christoffel symbols involves showing that the change in a vector along a curve can be calculated by subtracting the effect of parallel transport along the curve from the total change in the vector. This is done by using the Christoffel symbols to calculate the parallel transport term.

Geometrically, Christoffel symbols represent the connection between different tangent spaces in a curved space. They describe how vectors change as they move from one point to another on a curved surface.

Christoffel symbols are used in various fields of science, such as physics and engineering, to study the behavior of objects in curved spaces. They can be used to calculate the curvature of a space, solve differential equations, and make predictions about the behavior of physical systems. They are also used in the development of new technologies, such as space travel and GPS systems.

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