# Tensor (intrinsic Definition) - Tensor Fields

Tensor Fields

Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is in fact sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point.

### Other articles related to "tensor fields, tensor, tensors, tensor field, fields, field":

Mathematics Of General Relativity - Tensor Fields in General Relativity
... Tensor fields on a manifold are maps which attach a tensor to each point of the manifold ... bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle ... A tensor field is then defined as a map from the manifold to the tensor bundle, each point being associated with a tensor at ...
Levi-Civita Symbol - Tensor Density - Ordinary Tensor
... In the presence of a metric tensor field, one may define ordinary contravariant and contravariant tensor fields that agree with the Levi-Civita symbol wherever the coordinate system is such ... These ordinary tensor fields should not be confused with each other, nor should they be confused with the tensor density fields mentioned above ... One of these ordinary tensor fields may be converted to the other by raising or lowering the indices with the metric as is usual, but a minus sign is needed if the metric signature contains an odd number of negatives ...
Saint-Venant's Compatibility Condition - Rank 2 Tensor Fields
... The integrability condition takes the form of the vanishing of the Saint-Venant's tensor defined by The result that, on a simply connected domain W=0 ... are finite dimensional spaces of symmetric tensors with vanishing Saint-Venant's tensor that are not the symmetric derivative of a vector field ... Poincare's lemma can be understood more clearly using the operator, where is a symmetric tensor field ...

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