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Now i'm taking complex varibles course and learning about complex integration. But this is still beginning and i need to improve my knowledge about line integral. (I forgot a little about line integral and its parametrization on vector analysis, so i'm sorry if my question might be too easy for you guys.) And i have this problem.

To make it easy, I will numbering each equation so that you can call the number of the equation.

I wanna ask how to prove this equality, (Actually, in here, i'm using real line integral) :

$$\oint_C M(x,y)\,\mathbb ds=-\oint_C M(x,y)\,\mathbb ds\quad\tag{1}$$

It said, whenever we have a closed path no matter what path you choose, then the value of line integral is unchanged.

Even if it might not be relevant with complex line integral, i have a link that talks about that.

Line Integral

Then, in the second page on this page :

Properties of Line Integral

I'm starting to confuse with the second paragraph (proving the properties).

Cz it has another properties like :

$$\int_C M(x,y)\, \mathbb dx=-\int_C M(x,y)\,\mathbb dx\quad\tag 2$$

I don't know if i misunderstood or something. Is that mean on that form (general form and could be not a closed path) no matter the line integral is, we always have the same value in different direction?

Then what about

$$\int_C M(x,y)\, \mathbb dx=\int_{-C} M(x,y)\,\mathbb dx\quad\tag 3$$

and

$$\int_{-C} M(x,y)\, \mathbb dx=-\int_C M(x,y)\,\mathbb dx\quad\tag 4$$

And in the second page of link i found :

$$\int_{-C} M(x,y)\, \mathbb dx=-\int_{-C} M(x,y)\,\mathbb dx\quad\tag 5$$

All of those just make me more confused. I know, i need to learn more, but i can't find the best reference anymore for answer my question and prove the first equation. I've read journals, books and watching some videos on youtube, but i got nothing.


Edit :

What i'm trying to say is, How to prove :

$$\oint_{-C} M(x,y) \,\mathbb ds=\oint_{C} M(x,y) \,\mathbb ds$$

Please help.

Thanks in advance.

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This is similar to $$ \int _a^b f(x) dx = - \int _b ^a f(x)dx$$

When you trace a close curve in one direction you get the opposite of tracing the same curve on opposite direction.

Now, if instead of $dx$ you have $ds$ then the orientation does not matter because $ds$ is always positive.

That is $$\oint_C M(x,y)\,\mathbb ds=\oint_{-C} M(x,y)\,\mathbb ds\quad\tag{1}$$

But if one of the integrals is zero, then the other one is also zero so you get equality in a very special case.

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  • $\begingroup$ It helps me a lot, thanks. Anyway, for my first equation, is it allowed proving that equality with this? $\endgroup$ – user516076 Oct 12 '19 at 1:13
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    $\begingroup$ @user516076 the difference is with $ds$ verses $dx$. Note that with $ds$ the direction does not matter because $ds$ is positive with both orientations. $\endgroup$ – Mohammad Riazi-Kermani Oct 12 '19 at 1:19

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