FIELD

INDUCTION

NONLINEAR

LINEAR

DUAL

$\overline{D}{f}_{q+1}={\chi }_q\in T^{*}\otimes R_q$

$D{\xi }_{q+1}=X_q\in T^{*}\otimes R_q$

${\mathcal{X}}_q\in {\wedge }^n{T}^{*}\otimes T\otimes R_q^{*}$

${\chi }_{,r}^k$

${\chi }_{i,r}^k$

${\chi }_{r,i}^r$

${\chi }_{ij,r}^k$

${\partial }_i{\chi }_{r,j}^r-{\partial }_j{\chi }_{r,i}^r={\phi }_{ij}$

${\partial }_r{\xi }^k-{\xi }_r^k=X_{,r}^k$

${\partial }_r{\xi }_i^k-{\xi }_{i,r}^k=X_{i,r}^k$

${\partial }_i{\xi }_r^r-{\xi }_{ri}^r=X_{r,i}^r=X_i$

${\partial }_r{\xi }_{ij}^k-{\xi }_{ijr}^k={\partial }_r{\xi }_{ij}^k=X_{ij,r}^k$

${\partial }_i{\xi }_{r,j}^r-{\partial }_j{\xi }_{r,i}^r=F_{ij}$

$\frac{1}{2}\left({\partial }_i{\xi }_{r,j}^r+{\partial }_j{\xi }_{r,i}^r\right)=R_{ij}$

${\mathcal{X}}_k^{,r}$

${\mathcal{X}}_k^{i,r}$

${\mathcal{J}}^i$

${\mathcal{X}}_k^{ij,r}$

${\mathcal{F}}^{ij}$

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