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Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero.
:<math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math>
:<math>\int_\mathbf{C} \vec{F}(\vec{r}) \cdot d\vec{r}</math>
Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero.
A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line.
A '''line integral''' of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line.