Line Integral · aiwiki.page

A line integral is an integral taken along a curve rather than over an interval, area, or volume. In calculus, its two principal forms integrate either a scalar-valued function with respect to arc length or the tangential component of a vector field with respect to displacement. Despite its name, the curve need not be straight. Line integrals describe quantities such as the mass of a thin wire, work performed by a force, and circulation around a closed path. (openstax.org)

Curves and parametrization

A curve in Euclidean space can be represented by a parametrization

\mathbf r:[a,b]\longrightarrow\mathbb R^n.

The vector $\mathbf r(t)$ specifies position, while its derivative $\mathbf r'(t)$ specifies the tangent direction and rate of traversal. Elementary treatments generally use piecewise continuously differentiable curves, allowing corners at finitely many joining points. Increasing the parameter determines the curve’s orientation. (openstax.org)

Two differentials distinguish the main integrals:

ds=\|\mathbf r'(t)\|\,dt, \qquad d\mathbf r=\mathbf r'(t)\,dt.

Here $ds$ is a nonnegative element of arc length, and the norm measures speed. By contrast, $d\mathbf r$ is an oriented displacement. This distinction explains why reversing a curve leaves a scalar arc-length integral unchanged but reverses the sign of a vector work integral. (openstax.org)

Scalar line integrals

For a continuous function $f$ defined along a piecewise smooth curve $C$ , the scalar line integral is

\int_C f\,ds = \int_a^b f(\mathbf r(t))\|\mathbf r'(t)\|\,dt.

Its construction parallels a Riemann sum: divide the curve into short arcs, multiply a sampled function value by each arc’s length, and take the limit as the partition is refined. The speed factor is essential because equal parameter increments need not correspond to equal distances. (openstax.org)

Setting $f=1$ gives the length of $C$ . If $f=\lambda$ represents linear mass density, then

m=\int_C\lambda\,ds

gives the wire’s mass. Negative function values are permitted mathematically, although physical mass density is nonnegative. (openstax.org)

As a direct example, consider a circle of radius $R>0$ , traversed once:

\mathbf r(t)=(R\cos t,R\sin t),\qquad 0\le t\le2\pi.

Since its speed is $R$ , a constant density $\lambda_0$ gives

m=\int_0^{2\pi}\lambda_0R\,dt=2\pi R\lambda_0.

Vector line integrals and orientation

For a continuous vector field $\mathbf F$ , the oriented line integral is

\int_C\mathbf F\cdot d\mathbf r = \int_a^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt.

The dot denotes the Euclidean inner product. Equivalently, the integrand is $\mathbf F\cdot\mathbf T\,ds$ , where $\mathbf T$ is the oriented unit tangent. Thus only the component parallel to the curve contributes. In classical mechanics, integrating a force this way gives its work. (live.ocw.mit.edu)

In three dimensions, writing $\mathbf F=(P,Q,R)$ yields

\int_C\mathbf F\cdot d\mathbf r =\int_C P\,dx+Q\,dy+R\,dz.

A regular change of parameter preserving orientation leaves the value unchanged; reversing orientation changes its sign. Repeated traversal is not merely a change of parametrization: each additional traversal contributes again. (live.ocw.mit.edu)

Both principal forms are linear in the integrand and additive over successive curve segments. A closed-curve integral is commonly written $\oint_C$ ; for a vector field, it measures circulation. Unlike an arc-length integral, circulation can cancel because contributions have signs. (live.ocw.mit.edu)

Potentials and path independence

If $\mathbf F=\nabla\phi$ , where $\phi$ is continuously differentiable, the fundamental theorem for line integrals states

\int_C\nabla\phi\cdot d\mathbf r =\phi(B)-\phi(A),

with $A$ and $B$ the initial and final points. The gradient field therefore has a path-independent integral. The proof applies the chain rule to $\phi(\mathbf r(t))$ , followed by the fundamental theorem of calculus. (openstax.org)

On a connected open domain, a continuous field’s path independence is equivalent to its integral vanishing around every closed piecewise smooth path, and implies the existence of a scalar potential. For a conservative force, the usual mechanical convention is $\mathbf F=-\nabla U$ , so work equals the decrease in potential energy. (openstax.org)

Domain assumptions matter. A continuously differentiable field with zero curl is conservative on a simply connected open domain, but zero curl alone need not imply global path independence when the domain has holes. This is where topology enters the theory. (openstax.org)

Boundary theorems

Green’s theorem converts planar circulation into an area integral. If $C$ is a positively oriented, simple, closed, piecewise smooth boundary of a region $D$ , and $P,Q$ have continuous partial derivatives on a neighborhood of $D$ , then

\oint_C P\,dx+Q\,dy = \iint_D \left(\frac{\partial Q}{\partial x} -\frac{\partial P}{\partial y}\right)dA.

Its flux form instead relates the field’s outward normal component along the boundary to its divergence over the region. (openstax.org)

In three dimensions, Stokes’ theorem relates circulation around an oriented surface boundary to a surface integral of curl:

\oint_{\partial S}\mathbf F\cdot d\mathbf r = \iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS.

The boundary direction and surface normal must have compatible orientations. These identities permit a difficult curve integral to be replaced by an equivalent integral over a region or surface. (openstax.org)