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AiRAVATA

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Hello guys. Here I am, bothering all of you again...

I am having troubles proving the following:

For [itex]v \in V[/itex] and [itex]\omega \in \Lambda^k(V)[/itex], show that if [itex]v_1,v_2,...,v_n[/itex] is a basis of [itex]V[/itex] with dual basis [itex]\phi_1,\phi_2,...,\phi_n[/itex] then

[tex]i_{v_j}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})=\left\{ \begin{array}{ll} 0 & j\neq \hbox{ any } i_\alpha \\ (-1)^{\alpha-1} \phi_{i_1}\wedge ... \wedge \hat{\phi}_{i_\alpha}\wedge...\wedge \phi_{i_k} & \hbox{if }j=i_\alpha \right.[/tex]

where [itex]i_v \omega (v_1,...,v_{k-1})=\omega(v,v_1,...,v_{k-1})[/itex] and [itex]\hat{\phi}_{i_\alpha}[/itex] means we are omitting that term.

Use this result to show that for [itex]\omega_1 \in \Lambda^k(V)[/itex] and [itex]\omega_2\in \Lambda^l(V)[/itex] we have

[tex]i_{v}(\omega_1 \wedge \omega_2)=(i_v\omega_1)\wedge \omega_2+(-1)^k\omega_1\wedge(i_v\omega_2).[/tex]

Here is what I don't understand. In order to prove the first result, I apply the right side to [itex](v_1,...,v_{j-1},v_{j+1},...,v_k)[/itex], so

[tex]i_{v_j}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})(v_1,...,v_{j-1},v_{j+1},...,v_k)=(-1)^{j-1}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})(v_1,...,v_j,...,v_k)=\sum_{\sigma \in S} (-1)^\sigma \sigma (\phi_{i_1}\otimes ...\otimes\phi_{i_k}) (v_1,...,v_j,...,v_k)[/tex]

I'm stuck at this point. I know I need to evaluate the tensor product, but I'm going nowhere. Please someone enlighten me...

I am having troubles proving the following:

For [itex]v \in V[/itex] and [itex]\omega \in \Lambda^k(V)[/itex], show that if [itex]v_1,v_2,...,v_n[/itex] is a basis of [itex]V[/itex] with dual basis [itex]\phi_1,\phi_2,...,\phi_n[/itex] then

[tex]i_{v_j}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})=\left\{ \begin{array}{ll} 0 & j\neq \hbox{ any } i_\alpha \\ (-1)^{\alpha-1} \phi_{i_1}\wedge ... \wedge \hat{\phi}_{i_\alpha}\wedge...\wedge \phi_{i_k} & \hbox{if }j=i_\alpha \right.[/tex]

where [itex]i_v \omega (v_1,...,v_{k-1})=\omega(v,v_1,...,v_{k-1})[/itex] and [itex]\hat{\phi}_{i_\alpha}[/itex] means we are omitting that term.

Use this result to show that for [itex]\omega_1 \in \Lambda^k(V)[/itex] and [itex]\omega_2\in \Lambda^l(V)[/itex] we have

[tex]i_{v}(\omega_1 \wedge \omega_2)=(i_v\omega_1)\wedge \omega_2+(-1)^k\omega_1\wedge(i_v\omega_2).[/tex]

Here is what I don't understand. In order to prove the first result, I apply the right side to [itex](v_1,...,v_{j-1},v_{j+1},...,v_k)[/itex], so

[tex]i_{v_j}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})(v_1,...,v_{j-1},v_{j+1},...,v_k)=(-1)^{j-1}(\phi_{i_1}\wedge ... \wedge \phi_{i_k})(v_1,...,v_j,...,v_k)=\sum_{\sigma \in S} (-1)^\sigma \sigma (\phi_{i_1}\otimes ...\otimes\phi_{i_k}) (v_1,...,v_j,...,v_k)[/tex]

I'm stuck at this point. I know I need to evaluate the tensor product, but I'm going nowhere. Please someone enlighten me...

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