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Mathematics
Differential Geometry
Integration of differential forms
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[QUOTE="fresh_42, post: 6419261, member: 572553"] If I understand you correctly, you are asking about $$\displaystyle{\int_a^b \underbrace{\left(\omega\circ \gamma\right)\cdot \dfrac{d \gamma}{dt} }_{=:F(t)}\,dt}$$ and how length times height can be found in there, if we use the language of differential forms. You evaluated ##F(t)## at ##t## in that description, but refused to do the same for the [I]second[/I] [I]factor[/I] ##dt##. A covector is a function which assigns a real number to a vector, i.e. a direction. So if you [I]evaluate[/I] both, you get a product of real numbers summed up along the line from ##t=a## to ##t=b##. I'm not sure whether I like this picture or call it far fetched. One could as well interpret the ##dt## as pure symbolic sign meant to describe what runs in ##\int_a^b##. Under the keyword [I]exact [/I]you can find a specific example [URL='https://www.physicsforums.com/threads/why-the-terms-exterior-closed-exact.871875/#post-5474443']here[/URL], where I think the multiplications are more transparent. It is function times function and finally summed up over the points (of evaluation). [/QUOTE]
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Integration of differential forms
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