Line integral of scalar field ( piecewise curve)

[chetzread]

Homework Statement

for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ?

I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ?

Homework Equations

The Attempt at a Solution

is there anything wrong with the working ?

 

[drvrm]

for the line segment c2 , why did the author want to differentiate dx with respect to dy ? and he gt dx = 0 ?

actually one differentiates a function f(x,y) with respect to variable x or y;
your statement is in error - during c2 he is taking d/dy of f=x and as y is only variable on the path the slope of the path d/dy of x comes to zero.

I'm curious why did he didnt do so for C3 , where dy= 0 ..Why didnt he also differentiate dy with dx ? dy/dx = 0 ?

here also the author is taking d/dx of the function f=y as the variable is x ...not y as in the previous path and the slope is again zero.
slope of a curve is defined by tan of the angle made by a drawn tangent at the point under consideration..
Imagine f equivalent to say F(x,y) ...the graph of f is of a type x=0 for the c2 ( y from 1 to zero) and y=0 for c3 ( x going from 0 to 1)