For a continuous vector field \(\bm{F}\colon \mathbf{R}^3 \to \mathbf{R}^3\) defined in terms of its components as \(\bm{F} = L\mathbf{i} + M\mathbf{j} + N\mathbf{k}\) and a smooth curve \(C\) within that vector field with parameterization \(\bm{r}(t) = \bigl\langle x(t), y(t), z(t) \bigr\rangle\) for \(a \leq t \leq b,\) the line integral of \(\bm{F}\) along \(C\) is defined to be \[ \int_C \bm{F}\cdot\mathbf{T}\,\mathrm{d}s = \int_C \bm{F}\cdot\mathbf{T} \,\bigl({\color{maroon}\tfrac{\mathrm{d}s}{\mathrm{d}t}}\bigr)\,\mathrm{d}t = \int_a^b \bm{F}\bigl(\bm{r}(t)\bigr)\cdot\Bigl(\tfrac{\bm{r}'(t)}{|\bm{r}'(t)|}\Bigr) {\color{maroon} |\bm{r}'(t)| } \,\mathrm{d}t = \int_a^b \bm{F}\big(\bm{r}(t)\big)\cdot\bm{r}'(t)\,\mathrm{d}t = \int_C \bm{F}\cdot\mathrm{d}\bm{r} %= \int_C \bigl(L\mathbf{i}+M\mathbf{j}+N\mathbf{k}\bigr)\cdot\bigl(\tfrac{\mathrm{d}x}{\mathrm{d}t}\mathbf{i}+\tfrac{\mathrm{d}y}{\mathrm{d}t}\mathbf{j}+\tfrac{\mathrm{d}z}{\mathrm{d}t}\mathbf{k}\bigr) \,\mathrm{d}t = \int_C L \,\mathrm{d}x + M \,\mathrm{d}y + N \,\mathrm{d}z \] where \(\mathrm{d}s\) is a differential with respect to arclength and \(\mathbf{T}\) is the unit tangent vector at a point on \(C.\) If \(C\) is piecewise smooth, consisting of finitely many smooth segments, then the line integral of \(\bm{F}\) over \(C\) is the sum of the line integrals of \(\bm{F}\) over each segment. Note that per the orientation of \(C,\) \(\int_{-C} \bm{F}\cdot\mathrm{d}\bm{r} = -\int_{C} \bm{F}\cdot\mathrm{d}\bm{r}.\)
The Fundamental Theorem for Line Integrals, also called the Gradient Theorem — for a conservative vector field \(\bm{F}\) continuous on a smooth curve \(C\) with parameterization \(\bm{r}\) we have \[ \int_C \bm{F} \cdot \mathrm{d}\bm{r} = \int_C \nabla f \cdot \mathrm{d}\bm{r} = f\big(\bm{r}(b)\big) - f\big(\bm{r}(a)\big) \,. \] This is to say that line integrals in conservative vector fields are path independent: for a conservative vector field \(\bm{F}\) and any two curves (paths) \(C_1\) and \(C_2\) with coinciding initial and terminal points, \(\int_{C_1} \bm{F} \cdot \mathrm{d}\bm{r} = \int_{C_2} \bm{F} \cdot \mathrm{d}\bm{r}.\) In particular a line integral of a conservative vector field over a closed curve (loop) equals zero. This provides a separate characterization of conservative vector fields. Say \(R\) is a simply-connected region if it “has no holes”, if every closed curve in \(R\) can be contracted through \(R\) to a point in \(R.\) A vector field is conservative if it is path independent on any open connected region \(R,\) or if it’s mixed partials agree on any open simply-connected region \(R.\)