Covariant Derivative derivation.

In summary, the conversation discusses the use of the Leibniz rule and the equation \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b to show that \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b}. The conversation also mentions a scaler field \Phi and how it is introduced in the discussion. The conclusion is that the left hand side of the equation is the scaler field needed and everything works out nicely from there.
  • #1
T-chef
12
0

Homework Statement


Using the Leibniz rule and:
[tex] \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b [/tex]
[tex] \nabla_{a}\Phi=\partial\Phi [/tex]

Show that [itex] \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} [/itex].
The question is from Ray's Introducing Einsteins relativity,

My attempt:
[tex] \nabla_c(X^aX_a)=\nabla_c(X^a)X_a+X^a\nabla_c(X_a) [/tex]
[tex] = (\partial_{c}X^a+\Gamma_{bc}^{a}X^b)X_a+X^a\nabla_c(X_a) [/tex]

From here I'm not sure how to introduce the scaler field phi, or how doing so would help. Cheers for any help!
 
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  • #2
You already introduced it!
 
  • #3
Of course! The left hand side is exactly the scaler field I need. All comes out nicely after that! Thank you sir,
 

1. What is a covariant derivative?

A covariant derivative is a mathematical operation used in differential geometry to describe how the geometrical properties of a space change when one moves along a curve in that space. It takes into account the curvature and the metric of the space, making it a more general form of the ordinary derivative.

2. How is a covariant derivative different from an ordinary derivative?

An ordinary derivative measures the rate of change of a function with respect to one variable. A covariant derivative, on the other hand, takes into account the change in the function along a curve in a space, accounting for the curvature and metric of the space.

3. What is the purpose of deriving the covariant derivative?

The covariant derivative allows for the description of how vectors and tensors change as they are moved around in a curved space. It is a crucial tool in the study of differential geometry and is used in various fields such as general relativity and fluid dynamics.

4. How is the covariant derivative derived?

The covariant derivative is derived by introducing a connection, which is a way of connecting the tangent spaces at different points on a manifold. This connection allows for the calculation of the change of a vector or tensor along a curve, taking into account the curvature and metric of the space.

5. What are some applications of the covariant derivative?

The covariant derivative has many applications in physics, including general relativity, fluid dynamics, and electromagnetism. It is also used in differential geometry to study the properties of curved spaces. Additionally, it has applications in computer vision and image processing for feature detection and tracking.

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