性质与路径依赖性

The algebra any integral deserves

Do the endpoints alone determine the work

What path dependence means physically

Choosing parametrizations wisely

For $\mathbf{F} = (y, -x)$ from $(1,0)$ to $(0,1)$, the quarter circle gives $-\pi/2$ and the chord gives $-1$. What is the circulation around the loop they form?

When can you replace a line integral by a simpler path between the same endpoints?

A path runs from A to B with work $3$, then B to C with work $-5$. What is the work from A to C along the combined path?

A curve $C$ carries scalar line integral $7$ against arc length and vector line integral $4$. Give the vector line integral along the reversed curve, as a plain number.

For $\mathbf{F} = (y, -x)$ on the chord $\mathbf{r}(t) = (1-t, t)$, the velocity is $(-1, 1)$ and the dot product simplifies to a constant. Give that constant, as a plain integer like -1.