Exact differential of a scalar function f takes the form of(adsbygoogle = window.adsbygoogle || []).push({});

∇f⋅dr=Σ∂_{i}fdx_{i}(where dr is a vector)

f:R->R^{n}

and I am not sure why this equation is valid in the sense that if we integrate the equation,

∫∇f⋅dr=∫{Σ∂_{i}fdx_{i}}

∫df=∫{Σ∂_{i}fdx_{i}}

the above equation is true because integration is a linear operator, and if we think of R.H.S only,

∫{Σ∂_{1}fdx_{1}}+∫{Σ∂_{i≠1}fdx_{i≠1}}

=f+c(x_{2},x_{3}...,x_{n})+∫{Σ∂_{i≠1}fdx_{i≠1}}

and if we think of each integration of differential w.r.t variable x_{1},x_{2},x_{3}and so on, it should generate f+c respectively as the result of each integration.

therefore R.H.S has to be

nf+C(x_{1},x_{2},x_{3}....,x_{n}) but according to textbook, it says we dont add up each integration but we compare them to eradicate constants and "merge" each equation to one right answer, f.

I am a little bit confused about how to interpret the integration within the quotation mark ∫df="∫"{Σ∂_{i}fdx_{i}} because it seems it is linearly applied to each of partial differential, but does not spit out nf(n number of partial differentials so there must be n number of integrations on them so adding them up would give nf+C(x_{1},x_{2},x_{3}....,x_{n})

Please enlighten me. I would really appreciate

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# B Derivation of exact differential

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