Diagrammatic notation

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Diagrammatic notation is a (usually handwritten) notation of tensors commonly used by theoretical physicists. It consists of small shapes that have lines projecting upwards and downwards, representing abstract upper (contravariant) and lower (covariant) indices of tensors respectively.

Contents 1 Representation of special tensors 1.1 Metric tensor 1.2 Levi-Civita tensor 1.3 Structure constant 2 Tensor operations 2.1 Symmetrization 2.2 Antisymmetrization 2.3 Covariant derivative 3 Tensor manipulation

[edit] Representation of special tensors

[edit] Metric tensor

The metric tensor is represented by a U-shaped loop or an upside-down U-shaped loop, depending on the type of tensor that is used.

[edit] Levi-Civita tensor

The Levi-Civita antisymmetric tensor is represented by a thick horizontal bar with sticks pointing downwards or upwards, depending on the type of tensor that is used.

[edit] Structure constant

The structure constant () is represented by a small triangle with one line pointing upwards and two lines pointing downwards.

[edit] Tensor operations

[edit] Symmetrization

Symmetrization of indices is represented by a wavy line crossing horizontally across the index lines.

[edit] Antisymmetrization

Antisymmetrization of indices is represented by a thick bar crossing horizontally across the index lines.

[edit] Covariant derivative

The covariant derivative () is represented by a circle around the tensor(s) to be differentiated and a line joined from the circle pointing downwards to represent the lower index of the derivative.

[edit] Tensor manipulation

The diagrammatic notation is very useful in manipulating tensor algebra. It usually involves a few simple "identities" of tensor manipulations.

For example, , where n is the number of dimensions, is a common "identity".