Conservative Vector Fields — Question 8

Question 8 – Solution

Strategy. A closed curve can be compared with a constant path, while two endpoint-sharing paths form a closed curve when one is reversed.

Step 1: Path independence $\Rightarrow$ zero loops Let a closed curve start and end at $A$. Compare it with the constant path at $A$, whose line integral is zero. Path independence gives $$\oint_C\mathbf F\cdot d\mathbf r=0.$$

Step 2: Zero loops $\Rightarrow$ path independence Let $C_1$ and $C_2$ both run from $A$ to $B$. Follow $C_1$ from $A$ to $B$, then follow $-C_2$ back to $A$. This concatenation is closed.

By the assumed closed-loop condition, $$0=\int_{C_1}\mathbf F\cdot d\mathbf r +\int_{-C_2}\mathbf F\cdot d\mathbf r =\int_{C_1}\mathbf F\cdot d\mathbf r -\int_{C_2}\mathbf F\cdot d\mathbf r.$$ Therefore $$\boxed{\int_{C_1}\mathbf F\cdot d\mathbf r =\int_{C_2}\mathbf F\cdot d\mathbf r},$$ which is path independence.

Verification The reversal is essential: following $C_2$ in its original direction would begin at $A$, not at the terminal point $B$ of $C_1$, and would not close the loop.