For a continuous scalar field (multivariable function) \(f\colon \mathbf{R}^3 \to \mathbf{R}\) defined by a formula \(f(x,y,z)\) and a smooth curve \(C\) within that scalar field with parameterization \({\bm{r}(t) = \bigl\langle x(t), y(t), z(t) \bigr\rangle}\) for \({a \leq t \leq b,}\) the line (or path) integral of \(f\) on \(C\) is defined to be \[ \int_C f \,\mathrm{d}s = \int_C f \,\bigl({\color{maroon}\tfrac{\mathrm{d}s}{\mathrm{d}t}}\bigr) \, \mathrm{d}t = \int_a^b f\bigl(x(t), y(t), z(t)\bigr) {\color{maroon} \sqrt{\bigl(\tfrac{\mathrm{d}x}{\mathrm{d}t}\bigr)^2 \!\!+\! \bigl(\tfrac{\mathrm{d}y}{\mathrm{d}t}\bigr)^2 \!\!+\! \bigl(\tfrac{\mathrm{d}z}{\mathrm{d}t}\bigr)^2}} \,\mathrm{d}t = \int_a^b f\bigl(\bm{r}(t)\bigr) {\color{maroon}\bigl|\bm{r}'(t)\bigr|} \,\mathrm{d}t \] where \(\mathrm{d}s\) is a differential with respect to arclength. If \(C\) is piecewise smooth, consisting of finitely many smooth segments, then the line integral of \(f\) over \(C\) is the sum of the line integrals of \(f\) over each segment. If the curve \(C\) can be parameterized in terms of \(x\) or \(y\) or \(z\) then the line integral may be expressed with respect to that variable. \[ \int_C f \,\mathrm{d}x = \int_a^b \!f\bigl(x(t), y(t), z(t)\bigl) \,{\color{maroon}x'(t)} \,\mathrm{d}t \qquad \int_C f \,\mathrm{d}y = \int_a^b \!f\bigl(x(t), y(t), z(t)\bigl) \,{\color{maroon}y'(t)} \,\mathrm{d}t \qquad \int_C f \,\mathrm{d}z = \int_a^b \!f\bigl(x(t), y(t), z(t)\bigl) \,{\color{maroon}z'(t)} \,\mathrm{d}t \] When line integrals with respect to \(x\) and \(y\) and \(z\) occur together it’s conventional to write them as a single integral. E.g. \[ \int_C L \,\mathrm{d}x + \int_C M \,\mathrm{d}y + \int_C N \,\mathrm{d}z = \int_C L \,\mathrm{d}x + M \,\mathrm{d}y + N \,\mathrm{d}z \,. \]
A curve’s parameterization determines its orientation, the direction along which the curve is traversed. Given a curve \(C\) with implied orientation, we’ll let \(-C\) denote the same curve traversed in the opposite direction. A curve is simple if it doesn’t intersect itself, and a curve is closed if forms a loop. The Jordan Curve Theorem — a simple closed curve in the plane separates the plane into an interior and an exterior. Such a simple closed planar curve must be oriented either clockwise (left-handed) or counter-clockwise (right-handed).