I am given the integral: $$\int_{C} \frac{(y,-x)}{x^{2}+y^{2}} \cdot d \vec{C}$$ while$$C=\left\{(x, y) \mid x^{10}+y^{10}=100, x \leq 0\right\}$$ while $C$ is oriented downard.
The solution I have seen is :
We can check that $\vec{F}=\frac{(y,-x)}{x^{2}+y^{2}}$ is a conservative field, thus we can write $$\int_{C} \vec{F} \cdot d \vec{C}+\int_{C_{1}} \vec{F} \cdot d \vec{C}_{1}+\int_{C_{2}} \vec{F} \cdot d \vec{C}_{2}+\int_{C_{3}} \vec{F} \cdot d \vec{C}_{3}=0$$ while $C_{1}=\{x=0, y \in[-\sqrt[10]{100},-r]\}$ , $C_{2}=\{x=0, y \in[r, \sqrt[10]{100}]\}$ , $C_{3}=\left\{x^{2}+y^{2}=r^{2}, x \leq 0\right\}$
Because $\vec{F} \mid _{x=0}=\frac{(y, 0)}{y^{2}}$ we can conclude $\int_{C_{1}} \vec{F} \cdot d \vec{C}_{1}=0=\int_{C_{2}} \vec{F} \cdot d \vec{C}_{2}$
the rest of the solution is just calculation wich is not my goal right now.
My questions :
- how conservative force imply that $\int_{C} \vec{F} \cdot d \vec{C}+\int_{C_{1}} \vec{F} \cdot d \vec{C}_{1}+\int_{C_{2}} \vec{F} \cdot d \vec{C}_{2}+\int_{C_{3}} \vec{F} \cdot d \vec{C}_{3}=0$ where do I see the orientation of the domains he created?
- I draw the domain $C$ in desmos and I have no idea where the $C_3$ domain came from I just not understand how $C=C_1 +C_2 +C_3$
- where the $r$ came from in the domain of $C_1$ and $C_2$? what kind of substitute he did?