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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?

Alpha Delta
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1 Answers1

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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 $1\le i\le k$.

J.G.
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