Given a constant one-form [tex] k_1 \ dx + k_2 \ dy + k_3 \ dz [/tex] in [tex] \bold{R}^{3} [/tex], and three points [tex] \vec{a}, \ \vec{b}, \ \vec{c} [/tex] in [tex] \bold{R}^3 [/tex], prove that [tex] \int_{\vec{a}}^{\vec{c}} k_1 \ dx + k_2 \ dy + k_3 \ dz = \int_{\vec{a}}^{\vec{b}} k_1 \ dx + k_2 \ dy + k_3 \dz + \int_{\vec{b}}^{\vec{c}} k_1 \ dx + k_2 \ dy + k_3 \ dz [/tex].(adsbygoogle = window.adsbygoogle || []).push({});

So we want to show that [tex] k_{1}(c_1-a_1) + k_2(c_2-a_2) + k_3(c_3-a_3) = k_{1}(b_1-a_1) + k_2(b_2-a_2) + k_3(b_3-a_3) + k_1(c_1-b_1) + k_2(c_2-b_2) + k_3(c_3-a_3) [/tex].

Doesn't this follow from the transitive property, or the triangle inequality?

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# Proof of integral property

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