Welcome > Languages > Tensors

Tensors

Reference ¹

Vectors: ordered collection of numbers; one column matrices; a scalar magnitudes that have been given a direction, where direction is relative to the reference direction.
Vector space: two-space: (x,y); three space (x,y,z); Unit Vectors: $\hat x$, a magnitude of one.
Tensor: a mathematical representation of a physical entity that may be characterized by magnitude and multiple directions.
Rank: In a N-dimensional space, scalar has one number, vector will require N numbers, and R-rank tensor require N^R numbers.
- Scalar is tensors of rank 0;
- Vector is rank 1.
- 3 dimensional space: 2-rank tensor will have 9 numbers.
Component + Basic vectors:

Tensor:

1-rank in 3-dimension space: 3 1-rank basis, 3 components each vector;
2-rank in 3-dimension space: 9 2-rank basis, 9 components each vector; each component has two indices.
- Forces inside a solid object: for one point in the solid object, one surface can be defined by vector (xSurface,ySurface,zSurface), and the force can be defined as (xForce,yForce,zForce). Therefore, for this point, to represent all possible surfaces and all possible forces, we need a new coordinator system by combining all 3 axis of surfaces and 3 axis of forces. This results in a 9 axis space, each axis contain 1 surface axis and 1 force axis.

Tensor in greek: to stretch.

Stress tensor. rank-2 tensor.

can be written as 3x3 matrix, each represents a direction.

Index notation: rack n: need n indices for each vector component.

Coordinates transformation.

Tensors of rank one: (A_x, A_y, A_z): one index, or one basic vector per component. There is only one index for each of the vector component, because there is only one basis vector, (or directional indicator) for each compnent element.
Tensors of rank zero: need no index, because no directional indicator. The scalars.

If you could revise
the fundmental principles of
computer system design
to improve security...

... what would you change?