Penrose graphical notation

The penrosesche graphical notation - as penrosesche diagrammatic notation, tensor diagram notation or simply called Penrose notation - is a proposed by Roger Penrose notation in physics and mathematics, a (mostly handwritten ) visual representation of multi- linear maps or tensors receive. Is a diagram in this case consists closed molds which are connected via lines.

The notation has been extensively researched by Predrag Cvitanović, which uses this notation for the classification of classical Lie groups. The notation has been generalized to illustrate the theory of spin- networks in the physics and the presence of groups in the array of linear algebra.

Multi Linear Algebra

In the multilinear algebra corresponds to any form of multi-linear function. The lines of shapes represent the inputs or outputs of the function. The combination of these inputs and outputs corresponding to the composition of the respective functions.

Tensors

In the tensor algebra a certain tensor is shown as a particular form. Lines up and down abstract upper and lower indices of the respective tensors. Connections between two forms corresponds to the contraction of the indices. An advantage of this notation is that one does not have to invent new characters for new indexes. The notation is explicitly based independent.

Examples

Vector

Vector

Vector

Tensor

Tensor

Kronecker delta

Hermitian scalar product (see also: Bra- Ket )

Metric tensor

Metric tensor

Symmetrization

Asymmetrization

Tensor

( with )

Levi- Civita symbol ( permutation )

Faculty

Determinant

( Inverse of the matrix)

Trace function

Structure constant

Killing form

Covariant derivative

Riemannian curvature tensor

Ricci identity

Ricci tensor

Antisymmetry of the Riemann curvature tensor

Bianchi symmetry

Bianchi identity

(eg )