Covariant differentiation commutes with contraction?

In summary, the conversation discusses the use of the conclusion "covariant differentiation commutes with contraction" in tensor analysis, without proof. The conversation suggests two possible interpretations of this conclusion and recommends using the product rule and knowledge about \nabla^i \delta_k^m to prove it.
  • #1
kiddokiddo
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Homework Statement


I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.

Homework Equations


For example, [tex]\nabla_{i}T^{jk}_{kl}[/tex].

The Attempt at a Solution


I believe it can be interpreted in two ways. First, form the variant [tex]T^{jk}_{kl}[/tex] with two free indices j, l and apply [tex]\nabla_{i}[/tex] to that tensor; Or, apply [tex]\nabla_{i}[/tex] to the tensor [tex]T^{jk}_{ml}[/tex] and contract m and k. If the two interpretations lead to the same result, it should then be proved.

Any help is appreciated!
 
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  • #2
Whenever you contract something, there's a delta tensor involved. Try writing that out explicitly and using the product rule. What do you know about [itex]\nabla^i \delta_k^m[/itex]?
 

FAQ: Covariant differentiation commutes with contraction?

What is "Covariant Differentiation Commutes with Contraction"?

"Covariant Differentiation Commutes with Contraction" is a mathematical principle that states that the order in which a tensor is contracted and covariantly differentiated does not affect the final result. In other words, the result of the contraction and differentiation will be the same regardless of the order in which these operations are performed.

Why is this principle important in science?

This principle is important in science because it allows for a simpler and more efficient way of calculating equations involving tensors, which are commonly used in fields such as physics and engineering. It also helps to ensure the consistency and accuracy of mathematical calculations.

What are tensors?

Tensors are mathematical objects that describe the relationships between vectors and scalars in a multi-dimensional space. They are commonly used to represent physical quantities, such as velocity, force, and stress, in fields like physics and engineering.

How is this principle applied in real-world scenarios?

This principle is applied in various real-world scenarios, such as in the study of general relativity and fluid dynamics. In general relativity, tensors are used to describe the curvature of space-time, and the principle of covariant differentiation with contraction allows for simpler calculations of the equations of motion. In fluid dynamics, tensors are used to describe the properties of fluid flow, and the principle allows for efficient computation of equations governing the behavior of fluids.

Are there any limitations to this principle?

While the principle of covariant differentiation with contraction is generally applicable to most tensors, there are some cases where it may not hold true. For example, when dealing with non-linear transformations or non-constant coefficients, the principle may not apply. It is important to carefully consider the specific context and properties of the tensors being used before applying this principle.

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