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I know that while integrating dot product to two vector quantities along a line integral, the limits of the integration implicitly takes care of the direction in which we integrate from here and here.

But would this be true in case of cross products? Would the limits of the line integral of a cross product implicitly take care of the direction?

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  • $\begingroup$ Can you write the expression for cross-product (integral one which you are talking about)? $\endgroup$
    – Himanshu
    Commented Nov 19, 2020 at 7:18
  • $\begingroup$ @YoungKindaichi , It's a hypothetical question. I have not been able to think of an example yet. $\endgroup$
    – Alpha Delta
    Commented Nov 19, 2020 at 7:20
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    $\begingroup$ an example could be the rotational work done by an electric field on an electric dipole:$work = \int T d\Theta = \int ( \vec{p} \:\:\mathbf{x}\:\: \vec{E} )\:\:d\Theta $ $\endgroup$
    – lamplamp
    Commented Nov 19, 2020 at 7:30

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In general, limits on an integral over a subset of $\Bbb R^n$ implicitly take care of integration direction. (The case $n=1$ is familiar; if $a<b$ the directions for $\int_a^bfdx,\,\int_b^afdx$ are obvious.) It doesn't matter whether what's integrated is a univariate dot product, a $3$-dimensional cross product or in general a $k$-dimensional integrand, viz. $(\int_SVd^nx)_i=\int_SV_id^nx$ for $1\le i\le k$.

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