**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.

Read more about this topic: Tensor (intrinsic Definition)

### 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

... 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

... The integrability condition takes the form of the vanishing of the Saint-Venant's

**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**...### Famous quotes containing the word fields:

“And sweet it was to dream of Fatherland,

Of child, and wife, and slave; but evermore

Most weary seemed the sea, weary the oar,

Weary the wandering *fields* of barren foam.”

—Alfred Tennyson (1809–1892)

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