2階tensorの不変量の微分

$ \mathrm d\varphi=\varphi(\bm T+\mathrm d\bm T)-\varphi(\bm T)=\left.\frac{\mathrm d}{\mathrm d\epsilon}\varphi(\bm T+\epsilon\mathrm d\bm T)\right|_{\epsilon=0}

$ \mathrm d{\rm tr}\bm T=\left.\frac{\mathrm d}{\mathrm d\epsilon}{\rm tr}(\bm T+\epsilon\mathrm d\bm T)\right|_{\epsilon=0}={\rm tr}\mathrm d\bm T=\bm I:\mathrm d\bm T

$ \mathrm d{\rm tr}(\bm T^n)=\left.\frac{\mathrm d}{\mathrm d\epsilon}{\rm tr}((\bm T+\epsilon\mathrm d\bm T)^n)\right|_{\epsilon=0}=\left.{\rm tr}(n(\bm T+\epsilon\mathrm d\bm T)^{n-1}\cdot\mathrm d\bm T)\right|_{\epsilon=0}={\rm tr}(n\bm T^{n-1}\cdot\mathrm d\bm T)=n{\bm T^\top}^{n-1}:\mathrm d\bm T

$ \mathrm dI_1=\bm I:\mathrm d\bm T

$ \mathrm dJ_2=\frac12\mathrm d\left(I_1^2-{\rm tr}\left(\bm T^2\right)\right)

$ = \frac12(2I_1\bm I:\mathrm d\bm T-2\bm T^\top:\mathrm d\bm T)

$ = \bm T:(\bm I\bm I-\tilde{\cal\pmb I}):\mathrm d\bm T

$ \mathrm dI_3=I_3{\bm T^\top}^{-1}:\mathrm d\bm T

$ \mathrm d\lambda=\left.\frac{\mathrm d}{\mathrm d\epsilon}\lambda(\bm T+\epsilon\mathrm d\bm T)\right|_{\epsilon=0}

$ \det(\bm T+\epsilon\mathrm d\bm T-\lambda\bm I)=0