While proving the Stokes theorem in 2 dimension-
$$ \iint_R \left(\frac{\partial a_y}{\partial x }-\frac{\partial a_x}{\partial y}\right) dx\ dy = \oint_c(a_xdx +a_ydy) $$
In this book(Mathematical Methods For Physics And Engineering Textbook by K. F Riley), I came upon a step which seemed to be incorrect. First it took an arbitrary closed curve in 2 dimension, and defined the range of x and y coordinates. And started to solve the left side of the equation. The problem is this step-
$$ \int^{y_1}_{y_2} \left(\int_{x_1(y)}^{x_2(y)} \frac{\partial a_y}{\partial x }\ dx\right) dy=\int^{y_1}_{y_2}[a_y dy]_{x_1(y)}^{x_2(y)} $$
How can it directly convert $\frac{\partial a_y}{\partial x }dx$ to $da_y$ ?