If we have,$$A=d[(\bar{\alpha}-\alpha)(dt+\lambda)]$$(adsbygoogle = window.adsbygoogle || []).push({});

where $$\alpha$$ is a complex function and $$\lambda$$ is a 1-form. t here represents the time coordinate.

If we want to calculate $$d\star A=0$$ where $$\star$$ is hodge star, we get if I did my calculations correctly $$\nabla^2[(\bar{\alpha}-\alpha)(dt+\lambda)]=0$$

From the product rule, we get $$(\bar{\alpha}-\alpha)\nabla^2(dt+\lambda)+ (dt+\lambda)\nabla^2 (\bar{\alpha}-\alpha) +2 \nabla^2(\bar{\alpha}-\alpha)\nabla^2(dt+\lambda)=0$$

Does it mean here that each term must equal zero and thus since $$(\bar{\alpha}-\alpha)$$ can not be zero in the first term so $$\nabla^2(dt+\lambda)=0$$ thus killing with it the third term in the equation and only leaving the second term?

That is simplifying the above equation, we get, $$(dt+\lambda)\nabla^2 (\bar{\alpha}-\alpha)=0$$ thus leaving us with $$\nabla^2 (\bar{\alpha}-\alpha)=0$$

Can this step be done?

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# Hodge duality and differential forms

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