Show that the given integral is independent of a path. How do I do this?
Then, evaluate the line integral $I$ by finding a potential function $f$ and applying the fundamental theorem of line integrals.
Can someone please help I am so lost in this…
$$I = \int_{(0, 0)}^{(0, 2)} (x + y)\,dx + (x – y)\,dy$$
Best Answer
Define the function $f(x,y)=\frac{x^2}{2}+xy-\frac{y^2}{2}$. Prove that $\frac{\partial f}{\partial x}=x+y$ and $\frac{\partial f}{\partial y}=x-y$. In the language of your question, $f$ is a potential function.
Now, you want to integrate
$\int_{(0,0)}^{(0,2)} (x-y)dx+(x+y)dy$
$=\int_{(0,0)}^{(0,2)} \left(\frac{\partial f}{\partial x}\right)dx + \left(\frac{\partial f}{\partial y}\right)dy$
and this is always zero by the fundamental theorem of line integrals. (If you're still stuck, then look up the definition of "potential function" and the fundamental theorem of line integrals.)