INTEGRAIS COMPLEXAS E SUPERFÍCIES DE ANCELMO L. GRACELI [14]

integrais DE ANCELMO L. GRACELI.

INTEGRAIS DE ANCELMO L. GRACELI

X [ $\psi$ ] $\operatorname {Ric} _{p}$ = -S t [ $\psi$ ] $\operatorname {Ric} _{p}$ dt

X [ $\psi$ ] $\operatorname {Ric} _{p}$ = Z t [ $\psi$ ] [ $\operatorname {Ric} _{p}$ ] [ $e^{-{\frac {z}{\delta }}}$ ] dt

X [ $\psi$ ] $e^{-{\frac {z}{\delta }}}$ [ $\operatorname {Ric} _{p}$ ]= [ $\operatorname {Ric} _{p}$ ]Z t [ $\psi$ ] $\operatorname {Ric} _{p}$ [ $e^{-{\frac {z}{\delta }}}$ ] dt

X [ $\psi$ ] $\operatorname {Ric} _{p}$ = COS -S t [ $\psi$ ] $\operatorname {Ric} _{p}$ + SIN $\operatorname {Ric} _{p}$ [ $\psi$ ] d $\psi$

Ricci curvature tensor

It is a standard exercise of (multi)linear algebra to verify that this definition does not depend on the choice of the basis ⁠ $v_{1},\ldots ,v_{n}$ ⁠.

In abstract index notation, $\mathrm {Ric} _{ab}=\mathrm {R} ^{c}{}_{bca}=\mathrm {R} ^{c}{}_{acb}.$

Sign conventions. Note that some sources define $R(X,Y)Z$ to be what would here be called ⁠ $-R(X,Y)Z$ ⁠; they would then define $\operatorname {Ric} _{p}$ as ⁠ $-\operatorname {tr} (X\mapsto \operatorname {R} _{p}(X,Y)Z)$ ⁠. Although sign conventions differ about the Riemann tensor, they do not differ about the Ricci tensor.

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INTEGRAIS COMPLEXAS E SUPERFÍCIES DE ANCELMO L. GRACELI [14]

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