(adsbygoogle = window.adsbygoogle || []).push({}); Mistake in Carroll's Notes?

On the page 186 of Lecture Notes on General Relativity, where he writes the line element in both ##\tilde{u}## and ##\tilde{v}## coordinates , I think he missed a minus sign. Can someone check it, please?

Also, why he writes ##(d\tilde{u} d\tilde{v}+d\tilde{v}d\tilde{u})/2## instead of ##d\tilde{u} d\tilde{v}##?

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# Mistake in Carroll?

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