Discussion Overview
The discussion revolves around the definitions of the tensor double dot scalar product, specifically comparing two different formulations as they relate to tensor calculus and finite element methods. The scope includes theoretical aspects and potential applications in mathematical physics.
Discussion Character
- Debate/contested
- Technical explanation
Main Points Raised
- One participant presents two definitions of the tensor double dot scalar product: 1) \nabla \vec{u} \colon \nabla \vec{v} = u_{i,j} v_{j,i} and 2) \nabla \vec{u} \colon \nabla \vec{v} = u_{i,j} v_{i,j}, expressing a belief that the first is correct.
- Another participant mentions the outer multiplication between vectors and provides an index notation definition for the dyadic product.
- A participant notes that the first definition is more common but emphasizes that it is a matter of convention.
- One participant questions the validity of the two definitions, suggesting that since they yield different results, one must be correct and the other wrong.
- Another participant counters that the definitions are not wrong but simply different, providing examples of conventions that can lead to confusion in mathematical expressions.
Areas of Agreement / Disagreement
Participants express disagreement regarding which definition of the tensor double dot scalar product is correct, with some asserting that both are valid under different conventions. The discussion remains unresolved as to which definition should be preferred.
Contextual Notes
The discussion highlights the dependence on conventions in tensor calculus and the potential for confusion arising from different definitions. There are unresolved questions about the implications of using one definition over the other.