# Alt Tensor: If Alt(\omega)=\omega, Is \omega Alternating?

• yifli
In summary, "Alt Tensor" refers to the alternating tensor, a multilinear map used in mathematics and physics. In this context, "alternating" means that the map changes sign when the order of its inputs is switched. Alt(\omega) = \omega indicates that the differential form is invariant under the alternating map. The alternating map is used in various areas of mathematics, particularly in studying vector spaces, tensors, and defining integration and differentiation of differential forms. The significance of \omega being alternating lies in its properties that make it useful in mathematical calculations, as well as its relation to orientation in differential geometry.
yifli
If $\omega$ is an alternating tensor, then $Alt(\omega)=\omega$, where Alt is the mapping that maps any tensor to an alternating tensor.

I guess the converse is also true, i.e., if $Alt(\omega)=\omega$, then $\omega$ must be an alternating tensor. Am I right?

If you have w=Alt(w), and Alt(w) is an alternating tensor...

## 1. What does "Alt Tensor" mean?

"Alt Tensor" refers to the alternating tensor, which is a multilinear map that assigns a value of either 1 or -1 to each permutation of its inputs. It is used in mathematics and physics to study vector spaces, tensors, and differential forms.

## 2. What is the definition of "alternating" in this context?

In this context, "alternating" refers to a multilinear map that changes sign when the order of its inputs is switched. This means that if two inputs are switched, the resulting value will be the negative of the original value.

## 3. What does Alt(\omega) = \omega mean?

This notation means that the alternating map (Alt) applied to the differential form (\omega) will result in the same differential form. In other words, the differential form is invariant under the alternating map.

## 4. How is the alternating map used in mathematics?

The alternating map is used in various areas of mathematics, such as linear algebra, differential geometry, and differential equations. It is particularly useful in studying vector spaces and tensors, as well as in defining integration and differentiation of differential forms.

## 5. What is the significance of \omega being alternating?

The fact that \omega is alternating means that it satisfies certain properties that make it useful in mathematical calculations. For example, it allows for simplification of calculations involving tensors, and it is also related to the concept of orientation in differential geometry.

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