Line integral

A line integral is an integral computed along a curve in space, as opposed to an interval on the real line. Two flavors:

• Scalar line integral ${\int }_{C}fds$: integrate a scalar function, weighted by arc length.
• Vector line integral ${\int }_C\mathbf{F}\cdot d\mathbf{r}$: integrate a vector field, weighted by tangential displacement (dot product).

Both reduce to ordinary single-variable integrals once the curve is parameterized.

Scalar line integral

For a smooth curve $C$ parameterized by $\mathbf{r}(t)$, $a\le t\le b$, and a continuous scalar function $f$:

${\int }_{C}fds={\int }_a^{b}f(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|dt.$

Three steps: (i) evaluate $f$ along the curve, (ii) multiply by the speed $|{\mathbf{r}}^{\prime }(t)|$ (the arc-length element), (iii) integrate in $t$.

Independent of parameterization and orientation. Arc length is positive either way. So ${\int }_{C}fds$ sees only the shape of the curve.

Physical meaning. If $f$ is linear mass density of a curved wire, ${\int }_{C}fds$ is total mass. With $f\equiv 1$, you get arc length.

Vector line integral

For an oriented smooth curve $C$ and a continuous vector field $\mathbf{F}$:

${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_a^b\mathbf{F}(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt.$

Three steps: evaluate $\mathbf{F}$ at the curve, dot with the velocity, integrate.

Unit-tangent form. Using $\mathbf{T}={\mathbf{r}}^{\prime }/|{\mathbf{r}}^{\prime }|$ and $ds=|{\mathbf{r}}^{\prime }|dt$:

${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_C\mathbf{F}\cdot \mathbf{T}ds.$

Only the component of $\mathbf{F}$ along the direction of motion contributes. Perpendicular components contribute nothing.

Component form. Writing $\mathbf{F}=⟨P,Q,R⟩$ and $d\mathbf{r}=⟨dx,dy,dz⟩$:

${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_{C}Pdx+Qdy+Rdz.$

This is the form that appears in Green’s theorem and Stokes’ theorem.

Orientation matters. ${\int }_{-C}\mathbf{F}\cdot d\mathbf{r}=-{\int }_C\mathbf{F}\cdot d\mathbf{r}$ where $-C$ is $C$ in reverse. Unlike scalar line integrals.

Physical meanings of the vector line integral

• Work. If $\mathbf{F}$ is a force field, ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ is the work done on a particle moving along $C$. The dot product picks out the component of force in the direction of motion.
• Circulation. When $C$ is a closed curve, ${\oint }_C\mathbf{F}\cdot d\mathbf{r}$ is the circulation of $\mathbf{F}$ around $C$ — the net tendency of the field to flow around the loop.

“Circulation” is reserved for closed curves. For an open curve, ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ is a line integral or work integral, never a circulation. A circulation requires both a vector field and a closed loop — neither alone has a circulation.

Path dependence

In general, line integrals depend on the path, not just on the endpoints. Example from the textbook: $\mathbf{F}=⟨x^2-y,xy⟩$ from $(0,0)$ to $(2,4)$ gives $128/5$ along $y=x^2$, but $104/3$ along the L-shape (along $x$-axis then up). Same endpoints, different paths, different answers.

The exception is conservative (gradient) fields, where path-independence holds and ${\int }_C\mathbf{F}\cdot d\mathbf{r}=\varphi (B)-\varphi (A)$ for any path from $A$ to $B$.

Piecewise smooth curves

If $C$ has corners (like the L-shape), break it into smooth pieces and add the integrals.

Common parameterizations

• Line segment from $\mathbf{a}$ to $\mathbf{b}$: $\mathbf{r}(t)=(1-t)\mathbf{a}+t\mathbf{b}$, $0\le t\le 1$.
• Circle of radius $R$ in $xy$-plane, counterclockwise: $\mathbf{r}(t)=⟨R\cos t,R\sin t,0⟩$, $0\le t\le 2\pi$.
• Graph $y=g(x)$: $\mathbf{r}(t)=⟨t,g(t)⟩$.

In context

Line integrals are the building blocks for the central theorems of vector calculus:

• Fundamental theorem of line integrals: a gradient field’s line integral equals the potential difference.
• Green’s theorem: closed-loop circulation in 2D equals a double integral of curl.
• Stokes’ theorem: closed-loop circulation in 3D equals flux of curl.

A contour integral in complex analysis (Chapter 12 in Vector Calculus and Complex Analysis) is essentially a vector line integral in disguise — see Contour integral.