- #1

olgerm

Gold Member

- 472

- 25

I got another basic question: should the summation in einstein notation start from first occurance of index or in beginning of equation?

For eampledoes this equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }## from wikipedia mean:

## \sum_{j_1=0}^D(\sum_{j_2=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+{\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##

or

##\sum_{j_1=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+\sum_{j_2=0}^D({\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##?

For eampledoes this equation ##R_{\alpha \beta }={R^{\rho }}_{\alpha \rho \beta }=\partial _{\rho }{\Gamma ^{\rho }}_{\beta\alpha }-\partial _{\beta }{\Gamma ^{\rho }}_{\rho \alpha }+{\Gamma ^{\rho }}_{\rho \lambda }{\Gamma ^{\lambda }}_{\beta \alpha }-{\Gamma ^{\rho }}_{\beta \lambda }{\Gamma ^{\lambda }}_{\rho \alpha }## from wikipedia mean:

## \sum_{j_1=0}^D(\sum_{j_2=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+{\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##

or

##\sum_{j_1=0}^D(\frac{\partial{\Gamma ^{j_1}}_{\beta \alpha }}{\partial x^{j_1}}-\frac{{\Gamma^{j_1}}_{j_1 \alpha}}{\partial x^{\beta}}+\sum_{j_2=0}^D({\Gamma^{j_1}}_{j_1 j_2}{\Gamma ^{j_2}}_{\beta \alpha }-{\Gamma ^{j_1}}_{\beta j_2 }{\Gamma^{j_2}}_{j_1 \alpha }))##?

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