Line Integrals: Work Done by a Force and Path-Dependent Quantities

The cost of executing a trade depends not only on the start and end prices but on the path the price took — large orders experience market impact proportional to the sequence of transactions, not just the net displacement. This path-dependence is precisely what a vector line integral captures: the integral $\int_C\mathbf{F}\cdot d\mathbf{r}$ measures the accumulated "work" done by the force field $\mathbf{F}$ along the curve $C$. The central dichotomy — does the line integral depend on the path, or only on the endpoints? — will determine whether $\mathbf{F}$ is a conservative (gradient) field. This lesson defines scalar and vector line integrals, works through computations, and establishes the path-dependence vs. parameterisation-independence distinction.

Learning objectives. By the end of this note you should be able to: