Changes

:<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.  For a conservative field, the difference in potential between the endpoints of a curve equals the line integral along that same curve.

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 this is a [[conservative field]] and its 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.  For a conservative field, the difference in potential between the endpoints of a curve equals the line integral along that same curve.

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 curve.  Put another way, the line integral is the area under the function and along the curve.  The familiar, basic integral is simply the line-integral using the x-axis as the curve.

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 curve.  Put another way, the line integral is the area under the function and along the curve.  The familiar, basic integral is simply the line-integral using the x-axis as the curve.