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| <ref>SPENCER, A. J. M. Continuum Mechanics. Longman, 1980.</ref>{{Expert-subject|Mathematics|date=November 2008}}
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| In [[mathematics]], in the fields of [[multilinear algebra]] and [[representation theory]], '''invariants of tensors''' are coefficients of the [[characteristic polynomial]] of the [[tensor]] ''A'':
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| :<math>\ p(\lambda):=\det (\mathbf{A}-\lambda \mathbf{E})</math>,
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| where <math>\mathbf E</math> is the identity tensor and <math>\lambda\in\mathbb C</math> is the polynomial's [[Indeterminate (variable)|indeterminate]] (it is important to bear in mind that a polynomial's indeterminate <math>\lambda</math> may also be a non-scalar as long as power, scaling and adding make sense for it, e.g., <math>p(\mathbf A)</math> is legitimate, and in fact, quite useful).
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| The first invariant of an ''n''×''n'' tensor A (<math>I_A</math>) is the coefficient for <math>\lambda^{n-1}</math> (coefficient for <math> \lambda^n </math> is always 1), the second invariant (<math>II_A</math>) is the coefficient for <math>\lambda^{n-2}</math>, etc., the ''n''th invariant is the free term.
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| The definition of the ''invariants of tensors'' and specific notations used throughout the article were introduced into the field of [[Rheology]] by [[Ronald Rivlin]] and became extremely popular there. In fact even the [[Trace (linear algebra)|trace]] of a tensor <math>A</math> is usually denoted as <math>I_A</math> in the textbooks on rheology.
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| ==Properties==
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| The first invariant (trace) is always the sum of the diagonal components:
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| :<math>\ I_A=A_{11}+A_{22}+ \cdots + A_{nn}=\mathrm{tr}(\mathbf{A}) \, </math>
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| The ''n''th invariant is just <math>\pm \det \mathbf{A}</math>, the determinant of <math>\mathbf{A}</math> (up to sign).
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| The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective.
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| ==Calculation of the invariants of symmetric 3×3 tensors==
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| Most [[Application of tensor theory in engineering science|tensors used in engineering]] are symmetric 3×3.
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| For this case the invariants can be calculated as:
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| :<math>\mathrm{I}_A= \mathrm{tr}(\mathbf{A}) = A_{11}+A_{22}+A_{33} = A_1+A_2+A_3 \, </math>
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| :<math> | |
| \begin{align}
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| \mathrm{II}_A & = \frac{1}{2} \left( (\mathrm{tr}\mathbf{A})^2 - \mathrm{tr}(\mathbf{A} \mathbf{A}) \right) \\
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| & = A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}^2-A_{23}^2-A_{13}^2
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| \end{align}
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| </math>
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| (the sum of [[principal minor]]s)
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| :<math>
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| \begin{align}
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| & = A_1A_2+A_2A_3+A_1A_3 \\
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| \mathrm{III}_A & = \det (\mathbf{A})= A_1 A_2 A_3
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| \end{align}
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| </math>
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| where <math>A_1</math>, <math>A_2</math>, <math>A_3</math> are the [[eigenvalues]] of tensor ''A''.
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| Because of the [[Cayley–Hamilton theorem]] the following equation is always true:
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| :<math>\ \mathbf{A}^3 - \mathrm{I}_A \mathbf{A}^2 +\mathrm{II}_A \mathbf{A} -\mathrm{III}_A \mathbf{E}= 0</math>
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| where ''E'' is the second-order identity tensor.
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| A similar equation holds for tensors of higher order.
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| ==Engineering application==
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| A scalar valued tensor function ''f'' that depends merely on the three invariants of a symmetric 3×3 tensor <math>\mathbf{A}</math> is objective, i.e., independent from rotations of the coordinate system. Moreover, every objective tensor function depends only on the tensor's invariants. Thus, objectivity of a tensor function is fulfilled if, and only if, for some function <math>\ g:R^3\to R </math> we have
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| :<math>\ f(\mathbf A)=g(I_A,II_A,III_A). \, </math>
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| A common application to this is the evaluation of the potential energy as function of the strain tensor, within the framework of linear elasticity. Exhausting the above theorem the free energy of the system reduces to a function of 3 scalar parameters rather than 6. Within linear elasticity the free energy has to be quadratic in the tensor's elements, which eliminates an additional scalar. Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as Lame coefficients. Consequently, experimental fits and computational efforts may be eased significantly.
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| ==See also==
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| *[[Symmetric polynomial]]
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| *[[Elementary symmetric polynomial]]
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| *[[Newton's identities]]
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| *[[Invariant theory]]
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| ==References==
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| {{Unreferenced|date=October 2007}}
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| {{reflist}}
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| [[Category:Tensors]]
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| [[Category:Invariant theory]]
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| [[Category:Linear algebra]]
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