So, as i understand, the geometrical meaning of this type of integral should still be the area under the curve, however, I really do not see how you can obtain each infinitesimal rectangle from the dot product.(adsbygoogle = window.adsbygoogle || []).push({});

I have understood the typical work example, that is, the line integral as the sum of all the work done by each force in the direction of its respective displacement vector, but this only make sense in such a context, and not as a general mathematical conception.

However, influenced by such idea i tried to understand it as the influence of a specific field vector over each displacement vector that composes the curve, and given that the particle or object can not move outside of the curve's path, we ignore the parts of the field vector that are not in the same direction as the displacement vector, however (again) I do not see why would we define such influence as the product of the component of the field vector in the same direction as the displacement one with the displacement one itself (well their norms of course).

Can anyone help me out explaining the reasoning of this equation, specifically, why is the dot product in it?

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# General meaning of line integral in vector fields

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