Line and surface integrals · Definition

Line integral of a vector field

multivariable calculusline integrals

Before you start you need to know

It turns a force field and a path into a single number: the total work. It is the tool in which Green's, Stokes' and the divergence theorems are stated.

Definition

The line integral of a vector field \(\mathbf F\) over an oriented curve \(C\), parametrized by \(\mathbf r(t)\), \(a\le t\le b\), measures the work \(\mathbf F\) does along \(C\).

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

For a constant force the work is \(\mathbf F\cdot\Delta\mathbf r\). If the force varies and the path curves, split \(C\) into tiny pieces \(d\mathbf r=\mathbf r'(t)\,dt\) and add up the work \(\mathbf F\cdot d\mathbf r\) of each piece; that limit is the integral. Only the component of \(\mathbf F\) along the motion counts, which is why it is also written \(\int_C \mathbf F\cdot\mathbf T\,ds\), with \(\mathbf T\) the unit tangent vector. When \(C\) is closed it is called the circulation of \(\mathbf F\) around \(C\).

To compute it, parametrize \(C\), substitute into \(\mathbf F\), take the dot product with \(\mathbf r'(t)\) and integrate in \(t\). For example, with \(\mathbf F=\langle -y,x\rangle\) over the unit circle \(\mathbf r=\langle\cos t,\sin t\rangle\), the integrand is \(\sin^2t+\cos^2t=1\) and the circulation is \(2\pi\). In \(\mathbb R^3\) the procedure is identical with three components; if \(C\) has several pieces, add them.

Unlike the scalar line integral, orientation matters here: traversing \(C\) backwards flips the sign of \(\mathbf r'\) and therefore of the integral, so \(\int_{-C}\mathbf F\cdot d\mathbf r=-\int_C\mathbf F\cdot d\mathbf r\). Reparametrizing without reversing direction (going faster or slower) does not change the result, because the value depends on the curve and not on the speed.

Typical mistake: integrating \(\|\mathbf F\|\) or \(\mathbf F\) without taking the dot product with \(\mathbf r'\), or ignoring orientation when comparing two paths. This integral builds on vector fields and the scalar line integral, and raises the question of when the result depends only on the endpoints: the answer is conservative fields.

Think about it

How much work does \(\mathbf F=\langle 0,-9\rangle\) do on a particle moving from \((0,0)\) to \((5,2)\) in a straight line?

\(\mathbf F\cdot\Delta\mathbf r=0\cdot5+(-9)\cdot2=-18\): the field opposes the vertical motion.

If \(\mathbf F\) always points opposite to the direction of travel, what is the sign of the integral?

Negative, because \(\mathbf F\cdot\mathbf r'<0\) at every point.

Exercises 10

  1. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle y,x\rangle\) and \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
  2. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle -y,x\rangle\) over the unit circle traversed once counterclockwise.
  3. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle x,y,z\rangle\) and \(\mathbf r(t)=\langle t,t,t\rangle\), \(0\le t\le2\).
  4. If \(\int_C \mathbf F\cdot d\mathbf r=7\), what is the integral over the same curve traversed in the opposite direction, \(-C\)?
  5. A constant force \(\mathbf F=\langle2,3\rangle\) moves a particle in a straight line from \((0,0)\) to \((4,1)\). How much work does it do?
  6. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle xy,\,x\rangle\) and \(\mathbf r(t)=\langle t,t^2\rangle\), \(0\le t\le1\).
  7. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle z,x,y\rangle\) and \(\mathbf r(t)=\langle t,t^2,t^3\rangle\), \(0\le t\le1\).
  8. A field \(\mathbf F\) is perpendicular to the curve \(C\) at every point. What is \(\int_C \mathbf F\cdot d\mathbf r\)?
  9. The same curve \(C\) is traversed with the same orientation, but with a parametrization twice as fast. The line integral of \(\mathbf F\)…
  10. Compute \(\int_C \mathbf F\cdot d\mathbf r\) with \(\mathbf F=\langle 2xy,\,x^2\rangle\) and \(\mathbf r(t)=\langle\cos t,\sin t\rangle\), \(0\le t\le\tfrac\pi2\).

References

  • James Stewart, Daniel K. Clegg, Saleem Watson, Calculus: Early Transcendentals, 9th edition, Cengage, 2021. §16.2
  • Joel Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr., Thomas' Calculus, 15th edition, Pearson, 2023. §16.2
  • Jerrold E. Marsden, Anthony Tromba, Vector Calculus, 6th edition, W. H. Freeman, 2012. §7.2