Vector line integral weighted by a scalar field (${\int }_{C}Vd\mathbf{r}$)

For an oriented parametric curve $\mathbf{r}(t)$ and scalar field $V$, define the componentwise line integralIts value is a vector and reversing orientation changes its sign. It differs from the scalar arc length integral ${\int }_{C}Vds=\int V(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|dt$, which is independent of orientation. A request to integrate a scalar field along a curve should specify which differential is intended.

For the counterclockwise circle $\mathbf{r}(\varphi )=(R\cos \varphi ,R\sin \varphi ,z_0)$, suppose the restricted field has Fourier series $V(\varphi )=A_0+{\sum }_{n\ge 1}(A_n\cos n\varphi +B_n\sin n\varphi )$. Orthogonality givesThe oriented vector integral selects the first angular harmonics, whereas the scalar arc-length integral selects the mean.