Is the ordinary integral a special case of the line integral?

LucasGB

Can I consider the ordinary integral over the real line a special case of the line integral, where the line is straight and the field is defined only along the line?

HallsofIvy

You certainly can do it that way, if you want to.

LucasGB

But can it be called a field if it is only defined along a 1D line?

danzibr

If the field "defined" on the line actually satisfies the requirements to be a field, yes. Say, for example, if you're in C and choose your line to be R, then you're good. But if you choose the imaginary axis to be your line and do not change the definition of multiplication then you do not have a field.

But as for your question, I'm not sure. I think the two may be inherently different. If we're in R^2 the regular integral of any function along a line will be 0 because lines have measure 0, whereas it would be silly to have line integrals be 0 on all lines.