02 - tensor calculus tensor algebra
02 - tensor calculus
1
tensor calculus
tensor
the word tensor was introduced
in 1846 by william rowan hamilton. it was
used in its current meaning by woldemar
voigt in 1899. tensor calculus was developed around 1890 by gregorio ricci-curbastro under the title absolute differential
calculus. in the 20th century, the subject
came to be known as tensor analysis, and
achieved broader acceptance with the introduction of einsteins's theory of general
relativity around 1915. tensors are used
also in other fields such as continuum
mechanics.
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tensor calculus
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tensor calculus - repetition
• vector algebra
notation, euklidian vector space, scalar product,
vector product, scalar triple product
• tensor algebra
notation, scalar products, dyadic product, invariants,
trace, determinant, inverse, spectral decomposition,
sym-skew decomposition, vol-dev decomposition,
orthogonal
tensor
• tensor
analysis
derivatives, gradient, divergence, laplace operator,
integral transformations
tensor calculus
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vector algebra - notation
• einstein‘s summation convention
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• summation over any indices that appear twice in a
term
tensor calculus
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vector algebra - notation
• kronecker symbol
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• permutation symbol
tensor calculus
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vector algebra - euklidian vector space
• euklidian vector space
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•
is defined through the following axioms
• zero element and identity
• linear independence of
is the only (trivial)
solution to
if
tensor calculus
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vector algebra - euklidian vector space
• euklidian vector space
equipped with norm
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• norm defined through the following axioms
tensor calculus
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vector algebra - euklidian vector space
• euklidian vector space
euklidian norm
equipped with
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• representation of 3d vector
with
the
basis
relative
to
coordinates (components) of
tensor calculus
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vector algebra - scalar product
• euklidian norm enables definition of scalar (inner)
product
• properties of scalar product
• positive definiteness
• orthogonality
tensor calculus
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vector algebra - vector product
• vector product
• properties of vector product
tensor calculus
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vector algebra - scalar triple product
• scalar triple product
area
volume
• properties of scalar triple product
• linear independency
tensor calculus
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tensor algebra - second order tensors
• second order tensor
with
the
basis
relative
to
coordinates (components) of
• transpose of second order tensor
tensor calculus
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tensor algebra - second order tensors
• second order unit tensor in terms of kronecker
symbol
with
basis
the
coordinates (components) of
relative to
• matrix representation of coordinates
• identity
tensor calculus
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tensor algebra - third order tensors
• third order tensor
with
to the basis
relative
coordinates (components) of
• third order permutation tensor in terms of
symbol
permutation
tensor calculus
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tensor algebra - fourth order tensors
• fourth order tensor
with
coordinates (components) of
relative to the basis
• fourth order unit tensor
• transpose of fourth order unit tensor
tensor calculus
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tensor algebra - fourth order tensors
• symmetric fourth order unit tensor
• screw-symmetric fourth order unit tensor
• volumetric fourth order unit tensor
• deviatoric fourth order unit tensor
tensor calculus
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tensor algebra - scalar product
• scalar (inner) product
of second order tensor
and vector
• zero and identity
• positive definiteness
• properties of scalar product
tensor calculus
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tensor algebra - scalar product
• scalar (inner) product
of two second order tensors
and
• zero and identity
• properties of scalar product
tensor calculus
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tensor algebra - scalar product
• scalar (inner) product
of two second order tensors
• scalar (inner) product
of fourth order tensors
and second order tensor
• zero and identity
tensor calculus
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tensor algebra - dyadic product
• dyadic (outer) product
of two vectors
introduces second order tensor
• properties of dyadic product (tensor notation)
tensor calculus
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tensor algebra - dyadic product
• dyadic (outer) product
of two vectors
introduces second order tensor
• properties of dyadic product (index notation)
tensor calculus
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