Most tensors are formed from other tensors.
A scalar is a tensor of order 0. Examples include mass, energy, and entropy. These tensors are difficult to make. We find them in nature.
A vector is a tensor of order 1. Examples include displacement, force, electric field.
Start with tensors you know, you can make new tensors by recipes like these:
- A linear map that maps tensor 1 to tensor 2 is a new tensor.
- A bilinear map that maps tensor 1 and tensor 2 to tensor 3 is a new tensor.
You can invent your own recipes, so long as you start with known tensors and use linear maps.
Stress. By force balance, stress is a linear map that maps an area vector to the force acting on the area. See my notes on finite elasticity. The area vector is a tensor of order 1, and the force is a tensor of order 1. Consequently, the stress is a tensor of order 2.
Piezoelectricity. When a stress is applied on a piezoelectric material, an electric field arises. This phenomenon is modeled by a linear map that maps the stress to the electric field. The stress is a tensor of order 2, and the electric field is a tensor of order 1. Thus, the map is a tensor of order 3.
Please take a look at the recent thread on linear algebra and tensors.
Most tensors are formed from other tensors.
A scalar is a tensor of order 0. Examples include mass, energy, and entropy. These tensors are difficult to make. We find them in nature.
A vector is a tensor of order 1. Examples include displacement, force, electric field.
Start with tensors you know, you can make new tensors by recipes like these:
You can invent your own recipes, so long as you start with known tensors and use linear maps.
Stress. By force balance, stress is a linear map that maps an area vector to the force acting on the area. See my notes on finite elasticity. The area vector is a tensor of order 1, and the force is a tensor of order 1. Consequently, the stress is a tensor of order 2.
Piezoelectricity. When a stress is applied on a piezoelectric material, an electric field arises. This phenomenon is modeled by a linear map that maps the stress to the electric field. The stress is a tensor of order 2, and the electric field is a tensor of order 1. Thus, the map is a tensor of order 3.
Please take a look at the recent thread on linear algebra and tensors.