Transpose

2nd order tensors

Calculation

  • aijT=ajia_{ij}^{\mathrm{T}} = a_{ji} (If a=aije‾i⊗e‾j\boldsymbol{ a}=a_{ij}\underline{\boldsymbol{ e}}_{ i}\otimes\underline{\boldsymbol{ e}}_{ j}, then aT=aijTe‾i⊗e‾j\boldsymbol{ a}^{\mathrm{T}}=a_{ij}^{\mathrm{T}}\underline{\boldsymbol{ e}}_{ i}\otimes\underline{\boldsymbol{ e}}_{ j})

  • asym=0.5[a+aT]\boldsymbol{ a}^{ \mathrm{sym} } = 0.5\left[\boldsymbol{ a}+\boldsymbol{ a}^{\mathrm{T}}\right]

  • askw=a−asym\boldsymbol{ a}^{ \mathrm{skw} } = \boldsymbol{ a} - \boldsymbol{ a}^{ \mathrm{sym} }

Properties

  • [u‾⊗v‾]T=v‾⊗u‾\left[\underline{\boldsymbol{ u}}\otimes\underline{\boldsymbol{ v}}\right]^{\mathrm{T}} = \underline{\boldsymbol{ v}}\otimes\underline{\boldsymbol{ u}}

  • [ab]T=bTaT\left[\boldsymbol{ a}\boldsymbol{ b}\right]^{\mathrm{T}} = \boldsymbol{ b}^{\mathrm{T}}\boldsymbol{ a}^{\mathrm{T}}

  • [a+b]T=aT+bT\left[\boldsymbol{ a} + \boldsymbol{ b}\right]^{\mathrm{T}} = \boldsymbol{ a}^{\mathrm{T}} + \boldsymbol{ b}^{\mathrm{T}}

  • [aT]sym=asym\left[\boldsymbol{ a}^{\mathrm{T}}\right]^{ \mathrm{sym} } = \boldsymbol{ a}^{ \mathrm{sym} }

  • [aT]skw=−askw\left[\boldsymbol{ a}^{\mathrm{T}}\right]^{ \mathrm{skw} } = -\boldsymbol{ a}^{ \mathrm{skw} }

Differentiation

∂aT∂a=I⊗‾I\begin{aligned} \frac{\partial \boldsymbol{ a}^{\mathrm{T}}}{\partial \boldsymbol{ a}} = \boldsymbol{ I}\underline{\otimes}\boldsymbol{ I} \end{aligned}

4th order tensors (major transpose)

Calculation

  • AijklT= Aklij\textsf{ A}_{ ijkl}^{\mathrm{T}} = \textsf{ A}_{ klij}

  • Asym=0.5[ A+ AT]\textbf{\textsf{ A}}^{ \mathrm{sym} } = 0.5\left[\textbf{\textsf{ A}}+\textbf{\textsf{ A}}^{\mathrm{T}}\right]

  • Askw= A− Asym\textbf{\textsf{ A}}^{ \mathrm{skw} } = \textbf{\textsf{ A}} - \textbf{\textsf{ A}}^{ \mathrm{sym} }

Properties

  • [a⊗b]T=b⊗a\left[\boldsymbol{ a}\otimes\boldsymbol{ b}\right]^{\mathrm{T}} = \boldsymbol{ b}\otimes\boldsymbol{ a}

  • [ A: B]T= BT: AT\left[\textbf{\textsf{ A}}:\textbf{\textsf{ B}}\right]^{\mathrm{T}} = \textbf{\textsf{ B}}^{\mathrm{T}}:\textbf{\textsf{ A}}^{\mathrm{T}}

  • [ A+ B]T= AT+ BT\left[\textbf{\textsf{ A}} + \textbf{\textsf{ B}}\right]^{\mathrm{T}} = \textbf{\textsf{ A}}^{\mathrm{T}} + \textbf{\textsf{ B}}^{\mathrm{T}}

  • [ AT]sym= Asym\left[\textbf{\textsf{ A}}^{\mathrm{T}}\right]^{ \mathrm{sym} } = \textbf{\textsf{ A}}^{ \mathrm{sym} }

  • [ AT]skw=− Askw\left[\textbf{\textsf{ A}}^{\mathrm{T}}\right]^{ \mathrm{skw} } = -\textbf{\textsf{ A}}^{ \mathrm{skw} }

© Knut Andreas Meyer.
Last modified: April 22, 2024.
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