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}

Fundamental Theorem of Line Integrals

Suppose $C$ be a piecewise smooth curve parametrized by $\boldsymbol{r}(t)$ , $a \le t \le b$ .

If $\boldsymbol{F} = \nabla \phi$ on a region containing $C$ ,

\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = \int_C \nabla \phi \cdot \text{d}\boldsymbol{r} = \phi(\boldsymbol{r}(b)) - \phi(\boldsymbol{r}(a))

Worked Examples

Line Integral with Respect to Arc Length

Evaluate $\int_C xy\,\text{d}s$ , where $C$ is the quarter circle $\boldsymbol{r}(t) = \cos t\,\hat{i} + \sin t\,\hat{j}$ , $0 \le t \le \dfrac{\pi}{2}$ .

Speed:

\boldsymbol{r}'(t) = -\sin t\,\hat{i} + \cos t\,\hat{j}, \quad \lVert \boldsymbol{r}'(t) \rVert = \sqrt{\sin^2 t + \cos^2 t} = 1

Integral:

\int_C xy\,\text{d}s = \int_0^{\pi/2} \cos t \sin t \cdot 1\,\text{d}t = \left[\frac{\sin^2 t}{2}\right]_0^{\pi/2} = \frac{1}{2}

Integral Along a Parabola

Evaluate $\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r}$ for $\boldsymbol{F} = x^2\,\hat{i} + xy\,\hat{j}$ , where $C$ is $\boldsymbol{r}(t) = t\,\hat{i} + t^2\,\hat{j}$ , $0 \le t \le 1$ .

Field along the curve:

\boldsymbol{F}(\boldsymbol{r}(t)) = t^2\,\hat{i} + t^3\,\hat{j}, \quad \boldsymbol{r}'(t) = \hat{i} + 2t\,\hat{j}

Dot product:

\boldsymbol{F}(\boldsymbol{r}(t)) \cdot \boldsymbol{r}'(t) = t^2 + 2t^4

Integral:

\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = \int_0^1 (t^2 + 2t^4)\,\text{d}t = \frac{1}{3} + \frac{2}{5} = \frac{11}{15}

Work Along a Circle

Find the work done by $\boldsymbol{F} = -y\,\hat{i} + x\,\hat{j}$ on a particle moving once counterclockwise around the unit circle $\boldsymbol{r}(t) = \cos t\,\hat{i} + \sin t\,\hat{j}$ , $0 \le t \le 2\pi$ .

\boldsymbol{F}(\boldsymbol{r}(t)) = -\sin t\,\hat{i} + \cos t\,\hat{j}, \quad \boldsymbol{r}'(t) = -\sin t\,\hat{i} + \cos t\,\hat{j}

W = \int_0^{2\pi} (\sin^2 t + \cos^2 t)\,\text{d}t = \int_0^{2\pi} 1\,\text{d}t = 2\pi

Path-Independent Integral

Evaluate $\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r}$ for $\boldsymbol{F} = yz\,\hat{i} + xz\,\hat{j} + xy\,\hat{k}$ , where $C$ is any piecewise smooth curve from $(1,1,1)$ to $(2,3,4)$ .

Potential function:

\phi = xyz, \quad \nabla \phi = yz\,\hat{i} + xz\,\hat{j} + xy\,\hat{k} = \boldsymbol{F}

Integral:

\int_C \boldsymbol{F} \cdot \text{d}\boldsymbol{r} = \phi(2,3,4) - \phi(1,1,1) = 24 - 1 = 23