Define and give three examples of contravariant and covariant tensors ?
The Correct Answer and Explanation is :
In tensor analysis, contravariant and covariant tensors are fundamental concepts that describe how tensor components transform under changes in the coordinate system.
Contravariant Tensors:
Contravariant tensors have components that transform inversely to the transformation of the coordinate system. For example, if the coordinate system undergoes a scaling transformation, the components of a contravariant tensor will scale in the opposite direction. Vectors are a primary example of contravariant tensors. In Einstein notation, contravariant components are denoted with upper indices, such as ( v^i ). (Wikipedia)
Covariant Tensors:
Covariant tensors have components that transform directly with the coordinate system. For instance, under a scaling transformation, the components of a covariant tensor will scale in the same direction as the coordinate system. One example of a covariant tensor is the gradient of a scalar field, which can be represented as ( \nabla_i ) in Einstein notation. (Wikipedia)
Examples:
- Contravariant Vector:
- Position Vector: The position vector ( \mathbf{r} ) in three-dimensional space has components ( r^i ) that transform inversely to the coordinate system. For example, if the coordinate system is scaled by a factor of 2, the components of the position vector will scale by a factor of 1/2. (Wikipedia)
- Covariant Vector:
- Gradient of a Scalar Field: The gradient of a scalar field ( \phi ), denoted as ( \nabla_i \phi ), is a covariant vector. Under a scaling transformation of the coordinate system, the components of the gradient will scale in the same direction as the coordinate system. (Wikipedia)
- Contravariant Tensor:
- Metric Tensor: The metric tensor ( g^{ij} ) is a contravariant tensor that describes the geometry of space. Its components transform inversely to the coordinate system. (Wikipedia)
Understanding the distinction between contravariant and covariant tensors is crucial in fields like general relativity and differential geometry, as it influences how physical quantities are represented and transformed under changes in the coordinate system.
For a more in-depth explanation, you might find the following video helpful: