Math Prelim for tensors

Tensors & other Mathematical preliminaries

-Nilanjan Mitra [email protected]

www.facweb.iitkgp.ernet.in/~nilanjan

Cartesian coordinate frame: Positively oriented orthonormal basis

{e1, e2, e3}

Basis vectors obey

Inner product:

Cross product:

2

Relationship between Kronecker delta and permutation/alternating/LEVI-CIVITA symbol

Prove :

3

Tensor Product:

Prove :

4

Tensor

5

Look-back on inner, cross & tensor product of vectors (viewpoint of mapping)

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Tensor Algebra

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Tensor Algebra

Spherical and Deviatoric part of a Tensor

Inner product of a Tensor

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Determinant & cofactor of a Tensor

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Eigenvalue and eigenvectors

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Differentiation of functions of a scalar

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Differentiation of functions of a scalar

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Differentiation of functions of fields

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Divergence, Curl and Laplacian

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Differentiation of functions of fields

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Integral theorems

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