### Applet: The line integral of a path-dependent vector field

The vector field $\dlvf(x,y) = (y,-x)$ and three paths from $\vc{a}=(0,-4)$ (the cyan diamond) to $\vc{b}=(0,4)$ (the magenta diamond) as shown. Path $\dlc$ (shown in blue) is a straight line path from $\vc{a}$ to $\vc{b}$. Paths $\adlc$ (in green) and $\sadlc$ (in red) are clockwise and counterclockwise, respectively, circular path from $\vc{a}$ and to $\vc{b}$. You can drag the colored points, and the corresponding color lines on the slider indicate the line integral of $\dlvf$ from $\vc{a}$ along the path up to the colored point (the highlighted portion of the curve). Even when you drag the points to $\vc{b}$, the values of the line integrals have different values, demonstrating that $\dlvf$ is indeed path-dependent.

Applet file: line_integral_path_dependent_vector_field.ggb

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