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6: Line Integrals

Integrating functions along curves.

Line Integrals of Scalar Functions

Definition

For a curve CC parametrized by r(t)\mathbf{r}(t), a≤t≤ba \leq t \leq b:

∫Cf ds=∫abf(r(t))∣r′(t)∣ dt\int_C f \, ds = \int_a^b f(\mathbf{r}(t)) |\mathbf{r}'(t)| \, dt

where ds=∣r′(t)∣ dtds = |\mathbf{r}'(t)| \, dt is the arc length element.

Interpretation

  • If f=1f = 1: gives the arc length of CC
  • If f=ρf = \rho (density): gives the mass of a wire
  • Area of a “fence” along CC with height ff

Properties

  • Independent of parametrization direction
  • ∫Cf ds=∫C1f ds+∫C2f ds\int_C f \, ds = \int_{C_1} f \, ds + \int_{C_2} f \, ds for piecewise curves

Line Integrals of Vector Fields

Definition

For a vector field F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle along curve CC:

∫CF⋅dr=∫abF(r(t))⋅r′(t) dt\int_C \mathbf{F} \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt

Alternative Notations

∫CF⋅dr=∫CP dx+Q dy+R dz=∫CF⋅T ds\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C P \, dx + Q \, dy + R \, dz = \int_C \mathbf{F} \cdot \mathbf{T} \, ds

Physical Interpretation: Work

If F\mathbf{F} is a force field, ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r} is the work done moving along CC.

Properties

  • Depends on direction: ∫−CF⋅dr=−∫CF⋅dr\int_{-C} \mathbf{F} \cdot d\mathbf{r} = -\int_C \mathbf{F} \cdot d\mathbf{r}
  • Additive over paths: ∫C1+C2=∫C1+∫C2\int_{C_1 + C_2} = \int_{C_1} + \int_{C_2}

The Fundamental Theorem for Line Integrals

If F=∇f\mathbf{F} = \nabla f (conservative field) and CC goes from AA to BB:

∫C∇f⋅dr=f(B)−f(A)\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)

Key insight: For conservative fields, the line integral depends only on endpoints, not the path!

Conservative Vector Fields

A vector field F\mathbf{F} is conservative if F=∇f\mathbf{F} = \nabla f for some scalar function ff (called the potential function).

Equivalent Conditions

The following are equivalent for F\mathbf{F} on a simply connected domain:

  1. F\mathbf{F} is conservative (F=∇f\mathbf{F} = \nabla f)
  2. ∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r} is path-independent
  3. ∮CF⋅dr=0\oint_C \mathbf{F} \cdot d\mathbf{r} = 0 for every closed curve
  4. curl F=0\text{curl } \mathbf{F} = \mathbf{0}

Test for Conservative Field in 2D

F=⟨P,Q⟩\mathbf{F} = \langle P, Q \rangle is conservative iff:

∂P∂y=∂Q∂x\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}

Test for Conservative Field in 3D

F=⟨P,Q,R⟩\mathbf{F} = \langle P, Q, R \rangle is conservative iff:

∂P∂y=∂Q∂x,∂P∂z=∂R∂x,∂Q∂z=∂R∂y\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}, \quad \frac{\partial P}{\partial z} = \frac{\partial R}{\partial x}, \quad \frac{\partial Q}{\partial z} = \frac{\partial R}{\partial y}

Finding the Potential Function

If F=⟨P,Q⟩\mathbf{F} = \langle P, Q \rangle is conservative:

  1. Integrate: f=∫P dx=…+g(y)f = \int P \, dx = \ldots + g(y)
  2. Differentiate: ∂f∂y=Q\frac{\partial f}{\partial y} = Q
  3. Solve for g(y)g(y)

Applications

Work Done by a Force

W=∫CF⋅drW = \int_C \mathbf{F} \cdot d\mathbf{r}

Circulation

For a closed curve CC:

Circulation=∮CF⋅dr\text{Circulation} = \oint_C \mathbf{F} \cdot d\mathbf{r}

Measures the tendency of the field to circulate around CC.

Flux Across a Curve (2D)

Flux=∫CF⋅n ds\text{Flux} = \int_C \mathbf{F} \cdot \mathbf{n} \, ds

where n\mathbf{n} is the outward normal.

Summary

IntegralFormulaPhysical Meaning
Scalar line integral∫Cf ds\int_C f \, dsMass of wire, arc length
Vector line integral∫CF⋅dr\int_C \mathbf{F} \cdot d\mathbf{r}Work done by force
Conservative field∫C∇f⋅dr=f(B)−f(A)\int_C \nabla f \cdot d\mathbf{r} = f(B) - f(A)Path-independent
Circulation∮CF⋅dr\oint_C \mathbf{F} \cdot d\mathbf{r}Rotational tendency