Tensor & Matrix: Cartesian Vector & Transformation Rule?

  • Context: Graduate 
  • Thread starter Thread starter hokhani
  • Start date Start date
  • Tags Tags
    Matrix Tensor
Click For Summary
SUMMARY

The discussion centers on the relationship between Cartesian vectors and tensors, specifically addressing whether constant arrays such as ##a_{ij}; i,j=1,2,3## can be classified as components of a Cartesian tensor. It is established that while Cartesian refers to a coordinate system, tensors and vectors are inherently coordinate-independent. The representation of a tensor in a 3 x 3 Cartesian coordinate system is confirmed to be a matrix that adheres to the tensor transformation rule.

PREREQUISITES
  • Understanding of Cartesian coordinate systems
  • Familiarity with tensor transformation rules
  • Knowledge of matrix representation of tensors
  • Basic concepts of vector mathematics
NEXT STEPS
  • Research the properties of Cartesian tensors
  • Study the tensor transformation rules in detail
  • Explore the representation of tensors in different coordinate systems
  • Learn about the applications of tensors in physics and engineering
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are interested in the theoretical foundations and applications of tensors and their representations in Cartesian coordinate systems.

hokhani
Messages
601
Reaction score
22
Each set of constant numbers such as ##(v_1, v_2, v_3)## are the components of a constant Cartesian vector because by rotation of coordinates they satisfy the transformation rule. Can we consider each set of constant arrays ## a_{ij};i,j=1,2,3 ## as components of a Cartesian tensor? In other words, does each set of this type satisfy the tensor transformation rule?
 
Physics news on Phys.org
What do you mean by a Cartesian tensor? Cartesian is usually used to refer to a coordinate system, not a tensor, or a vector, both of which are coordinate-independent objects.
Given any 3 x 3 Cartesian coordinate system, the matrix you mention will be the representation in that coordinate system of a tensor.
 
andrewkirk said:
What do you mean by a Cartesian tensor? Cartesian is usually used to refer to a coordinate system, not a tensor, or a vector, both of which are coordinate-independent objects.
Given any 3 x 3 Cartesian coordinate system, the matrix you mention will be the representation in that coordinate system of a tensor.
Thanks. By "Cartesian tensor" I meant the representation of a tensor in Cartesian coordinate system.
 

Similar threads

Graduate Contravariance, Covariance, stress and position vector
  • · Replies 24 ·
Replies
24
Views
3K
High School Transformation Rules For A General Tensor M
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How to apply tensor transformation rule
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad Time derivative of the moment of inertia tensor
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Transformation of second rank tensor
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Derivation of equations of stress tensor transformation
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad A little clarification on Cartesian tensor notation
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad How do Tensors "work" in relation to linear algebraic objects?
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad Transforming coordinates between Cartesian and spherical
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How to calculate basis vectors from metric tensor (as a matrix)?
  • · Replies 6 ·
Replies
6
Views
1K