Covariant Derivative: Deriving the Equation

In summary, the conversation is about the covariant derivative and how it is defined. The speaker suggests searching for the term on Google for more information and clarifies that the discussion is taking place in the 'Calculus and Analysis' section of a mathematics forum. They also express frustration with the other person's brief and unclear responses, and provide a suggested rephrasing of the original question.
  • #1
John_Doe
51
0
How is the covariant derivative derived?
 
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  • #2
You mean, "How is the covariant derivative defined?", don't you?

Step 1: Open an internet browser.
Step 2: Go to Google.
Step 3: Search for "covariant derivative".

You will see entries from both Wikipedia and MathWorld.
 
  • #3
No. This thread has been continued in 'Calculus and Analysis' under 'Mathematics'.
 
  • #4
If you weren't so terse in your replies and actualy explained fully what you meant then you might get somewhere more quickly. A more careful choice of words would help as well. You don't mean derive, closer is why on taking the limit of something along certain paths or whatever this is (trying too recall prevect's post in the calc thread) is it equivalent to this statement? How about "how does one pass from the phrase 'is the limit along some parallel transport of something' to that formula?" That to my mind isn't deriving, it is simply moving from words to symbols. One derives the fundamental theorem of calculus by starting with the axioms and definitions of continuity etc and proving the result.
 
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Related to Covariant Derivative: Deriving the Equation

1. What is a covariant derivative?

A covariant derivative is a mathematical tool used in differential geometry to measure the change of a vector field along a given direction on a curved manifold. It takes into account the curvature of the manifold and allows for the differentiation of vector fields that are not parallel.

2. How is the covariant derivative equation derived?

The covariant derivative equation is derived by considering how the components of a vector field change along a given direction on a curved manifold. It takes into account the Christoffel symbols, which represent the curvature of the manifold, and the partial derivatives of the vector field components.

3. What is the importance of the covariant derivative equation?

The covariant derivative equation is important in understanding the behavior of vector fields on curved manifolds. It allows for the calculation of derivatives and gradients in a way that is consistent with the curvature of the manifold, which is necessary for many physical and mathematical applications.

4. Can the covariant derivative equation be extended to higher dimensions?

Yes, the covariant derivative equation can be extended to higher dimensions, known as multidimensional covariant derivatives. This is necessary for understanding more complex manifolds with higher dimensions, such as those found in general relativity.

5. How is the covariant derivative equation used in physics?

The covariant derivative equation is used in many areas of physics, particularly in the fields of general relativity, electromagnetism, and quantum mechanics. It is used to describe the behavior of fields and particles in curved spacetime and is essential in understanding the fundamental laws of physics.

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