Covariant Derivative: Limits on Making a Tensor?

Click For Summary
SUMMARY

The discussion centers on the limitations of covariant derivatives in relation to non-invariant quantities and tensors. It is established that covariant derivatives can only be applied to tensors, not to individual components. Therefore, one cannot derive an invariant tensor quantity from non-invariant components using covariant derivatives. This highlights the fundamental nature of tensors in differential geometry.

PREREQUISITES
  • Tensor calculus
  • Understanding of covariant derivatives
  • Knowledge of invariant quantities in differential geometry
  • Familiarity with the properties of tensors
NEXT STEPS
  • Study the properties of covariant derivatives in detail
  • Explore the relationship between tensors and invariant quantities
  • Learn about the applications of tensors in physics
  • Investigate the implications of tensor calculus in general relativity
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics and physics, particularly those focusing on differential geometry and tensor analysis.

dsaun777
Messages
296
Reaction score
39
Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity? Or are there limits on what you can make a tensor?
 
Physics news on Phys.org
dsaun777 said:
Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity?

You can't take covariant derivatives of components. Covariant derivatives operate on tensors.
 
PeterDonis said:
You can't take covariant derivatives of components. Covariant derivatives operate on tensors.
Sorry my brain is not working long day...
 

Similar threads

High School Prove that metric tensor is covariant constant
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Commutation between covariant derivative and metric
  • · Replies 16 ·
Replies
16
Views
4K
Graduate Proof of covariant derivative of spinor
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Ricci Tensor: Covariant Derivative & Its Significance
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Dirac comment on covariant derivatives
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Covariant divergence of vector; physical meaning with contracted Tuv
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Physical Difference Between Co- and Contravariant Vectors
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad What is a Covariant Relation? GR Problem Solution Explained
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Can We Explain the Cosmological Constant as a Tensorial Concept?
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates
  • · Replies 5 ·
Replies
5
Views
3K