Evaluate $\int_C x dz$, where $C$ is the union of the line segments joining $0$ to $i$, and then to $i + 2$.
In the solutions, it says $C = C_1 + C_2$, where:
$C_1: [0, 1] \rightarrow \mathbb{C}$, and $C_1 (y) = iy$, and
$C_2: [0, 2] \rightarrow \mathbb{C}$, and $C_2 (x) = x + i$.
I don't quite see how they arrived at these formulas for $C_1$ and $C_2$. I understand how we can break down $C$ into these 2 parts, but the actual "evaluations" of $C_1$ and $C_2$ confuses me. Why is it that $C_1$ is in terms of $y$ while $C_2$ is in terms of $x$? How did they arrive at these functions for both lines?
Any clarification would be greatly helpful. Thank you.
$x$ is being used to denote the real part of $z$ and $y$ the imaginary part.
$C_1$ is supposed to be the line segment joining $0$ to $i$. Think about what's happening to the real and imaginary parts of $z$ as it moves from $0$ to $i$. The real part stays constant at $0$, while the imaginary part (coefficient of $i$) goes up from $0$ to $1$. So the variable that is changing here is the imaginary part $y$ while the real part remains constant at $0$: $z = 0 + yi$.
On $C_2$, the imaginary part remains constant at $1$, while the real part goes from $0$ to $2$. So $z = x + 1i$, with $x$ going from $0$ to $2$.