Proving path independence of the sum of two path independent line integrals.

Question

How can we prove that if $$\int_{P_{1}}^{P_{2}} \overrightarrow{f_{1}} . \overrightarrow{dl}$$

and $$\int_{P_{1}}^{P_{2}} \overrightarrow{f_{2}} . \overrightarrow{dl}$$

are both path independent, i.e., only depend on the initial and final positions $P_{1}$ and $P_{2}$ , respectively, then

$$\int_{P_{1}}^{P_{2}} \overrightarrow{f_{1}} . \overrightarrow{dl}+\int_{P_{1}}^{P_{2}} \overrightarrow{f_{2}} . \overrightarrow{dl}$$

is path independent as well?

Answer 1

Hint: Write this sum along two arbitrary paths, and consider the difference.

Answer 2

Sum of two continuous functions is continuous. Since both integrals are path independent, $f_1$ and $f_2$ are continuous. By linearity if integral, we can define $f_3:=f_1+f_2$ and its line integral is path independent.