Line Integral | Sahithyan's Notes

Let $C$ be a smooth curve with parametric equations $x = x(t)$ , $y = y(t)$ , $z = z(t)$ , $a \le t \le b$ , and $f$ defined on $C$ . Let $P = \{t_0, t_1, \dots, t_n\}$ , $a = t_0 < t_1 < \dots < t_n = b$ , be a partition of $[a,b]$ , dividing $C$ into $n$ subarcs. Let $\Delta s_k$ be the length of the $k$ th subarc, and $(x_k^*, y_k^*, z_k^*)$ a sample point on the $k$ th subarc. The line integral of $f$ over $C$ with respect to arc length is

\int_C f(x,y,z)\,\text{d}s = \lim_{\max\{\Delta s_k\} \to 0} \sum_{k=1}^{n} f(x_k^*, y_k^*, z_k^*)\,\Delta s_k

provided the limit exists and does not depend on the choice of partition or sample points.

For $\boldsymbol{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$ , $a \le t \le b$ ,

\text{d}s = \lVert \boldsymbol{r}'(t) \rVert\,\text{d}t = \sqrt{\left(\frac{\text{d}x}{\text{d}t}\right)^2 + \left(\frac{\text{d}y}{\text{d}t}\right)^2 + \left(\frac{\text{d}z}{\text{d}t}\right)^2}\,\text{d}t

\int_C f(x,y,z)\,\text{d}s = \int_a^b f(x(t),y(t),z(t))\,\lVert \boldsymbol{r}'(t) \rVert\,\text{d}t

Line Integral of a Vector Field

Let $\boldsymbol{F}$ be a vector field and $C$ a piecewise smooth oriented curve parametrized by $\boldsymbol{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$ , $a \le t \le b$ . The line integral of $\boldsymbol{F}$ along $C$ is

\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = \int_a^b \boldsymbol{F}(\boldsymbol{r}(t)) \cdot \boldsymbol{r}'(t)\,\text{d}t

For $\boldsymbol{F} = P\hat{i} + Q\hat{j} + R\hat{k}$ ,

\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = \int_C P\,\text{d}x + Q\,\text{d}y + R\,\text{d}z

Work

For a particle moving along a smooth curve $C$ , oriented in the direction of motion, under a continuous force field $\boldsymbol{F}$ , the work done by $\boldsymbol{F}$ on the particle is

W = \int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r}