An exterior-algebraic derivation of the symmetric stress–energy–momentum tensor in flat space–time

Tensor, Exterior and Symmetric Algebras

3 The Exterior Algebra 6 3.1 Dimension of the Exterior Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Other Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3.1 The determinant formula . . . . . . . . . . . . . . . . . . . . . . . . . ...

Tensor Algebras, Symmetric Algebras and Exterior Algebras

We begin by defining tensor products of vector spaces over a field and then we investigate some basic properties of these tensors, in particular the existence of bases and duality. After this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Tensor products of modules over a commutative ring with identity will be discussed very briefly. They show up...

Lecture III: Tensor calculus and electrodynamics in flat spacetime

, we can define the outer product (or “tensor product”) to be the tensor A⊗ B with (A⊗B)(k̃, ...u, l̃, ...v) = A(k̃, ...u)B(l̃, ...v). (1) It is trivial to see that this is in fact a tensor – it is linear in each input. Its components are (A⊗B)1M β1...βN γ1...γP δ1...δQ = A1M β1...βNBγ...γ δ1...δQ . (2) This is the generalization to tensors of arbitrary rank of taking two column vectors u and v and...

Exterior and Symmetric Algebra Methods in Algebraic Combinatorics

A few classical results on tensor, symmetric and exterior powers

0.1. Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.3. Basic conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.4. Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.5. Tensor powers of k-modules . . . . . . . . . . . ...