tensor algebra वाक्य
उदाहरण वाक्य
मोबाइल
- The construction of the bialgebra here parallels the construction in the tensor algebra article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.
- They give rise to a " tensor " only by identifications with objects of the tensor algebra-but likewise there are identifications with concepts in the Clifford-algebra.
- The free algebra on a set " X " is for practical purposes the same as the tensor algebra on the vector space with " X " as basis.
- Conversely, symmetric tensors are defined as invariants : given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially.
- There is a general relativity book-that appears to be about half-finished-but it looks too complex for 14-16 year olds because it goes into tensor algebra and such.
- All the algebraic relations between the basis vectors, as discussed in the section on tensor algebra, apply for the natural basis and its reciprocal at each point "'x " '.
- Another example is when T is the endofunctor on the category of vector spaces which maps a vector space V to its tensor algebra T ( V ), and which maps linear maps to their tensor product.
- Where the sum is taken over the symmetric group of permutations on the symbols This extends by linearity and homogeneity to an operation, also denoted by Alt, on the full tensor algebra T ( " V " ).
- One can then show that the Lie bracket can be consistently lifted to the entire tensor algebra : it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted.
- The tensor product on the tensor algebra induces a product K _ m \ times K _ n \ rightarrow K _ { m + n } making K ^ M _ * ( F ) a graded ring which is graded-commutative.
- The Lie derivative with respect to a vector field is an "'R "'- derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold.
- We then have a natural transformations corresponding to the embedding of V into its tensor algebra, and a natural transformation corresponding to the map from T ( T ( V ) ) to T ( V ) obtained by simply expanding all tensor products.
- These two are sufficient to extend the notion of the Lie bracket to the entire tensor algebra, by appealing to a lemma : since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra.
- These two are sufficient to extend the notion of the Lie bracket to the entire tensor algebra, by appealing to a lemma : since the tensor algebra is a free algebra, any homomorphism on its generating set can be extended to the entire algebra.
- Here, the tensor product symbol ?" is used to indicate the tensor product used to define a coalgebra; it must not be confused with the tensor product ?", which is used to define the bilinear multiplication operator of the tensor algebra.
- There are other general examples, as well : it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article.
- Then contraction operates on the full ( mixed ) tensor algebra of " M " in exactly the same way as it does in the case of vector spaces over a field . ( The key fact is that the natural pairing is still perfect in this case .)
- It is being used to distinguish it from the " internal " tensor product \ otimes, which is already " taken " and being used to denote multiplication in the tensor algebra ( see the section " Multiplication ", below, for further clarification on this issue ).
- One can, in fact, define the tensor algebra " T " ( " V " ) as the unique algebra satisfying this property ( specifically, it is unique up to a unique isomorphism ), but one must still prove that an object satisfying this property exists.
- Because of the generality of the tensor algebra, many other algebras of interest can be constructed by starting with the tensor algebra and then imposing certain relations on the generators, i . e . by constructing certain quotient algebras of " T " ( " V " ).
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