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.
Then, in the second page on this page :
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.
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.