2.4 Line and Surface Integrals

Line integrals and surface integrals are powerful tools for analyzing scalar and vector fields along curves and surfaces. They allow us to calculate quantities like work done by forces or flux through surfaces, essential in physics and engineering applications.

Green's theorem bridges line and surface integrals, connecting the circulation of a vector field around a closed curve to the curl of the field over the enclosed region. This relationship simplifies calculations and provides insights into conservative fields and path independence.

Line Integrals

Scalar vs vector line integrals

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• Scalar line integrals integrate a scalar function along a curve (temperature along a wire)
  • Denoted as $\int_C f(x, y) ds$ where $ds$ represents the arc length element
• Vector line integrals integrate a vector field along a curve (work done by a force field)
  • Denoted as $\int_C \mathbf{F}(x, y) \cdot d\mathbf{r}$ where:
    • $\mathbf{F}(x, y)$ is a vector field (force field)
    • $d\mathbf{r}$ is the infinitesimal displacement vector along the curve (path)

Evaluation of line integrals

• Parametrization of a curve represents the curve as a function of a parameter $t$ (time)
  • $x = x(t)$, $y = y(t)$, where $a \leq t \leq b$
• Scalar line integral with parametrization:
  • $\int_C f(x, y) ds = \int_a^b f(x(t), y(t)) \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$
• Vector line integral with parametrization:
  • $\int_C \mathbf{F}(x, y) \cdot d\mathbf{r} = \int_a^b \mathbf{F}(x(t), y(t)) \cdot \left(\frac{dx}{dt}, \frac{dy}{dt}\right) dt$
• Fundamental theorem of line integrals states that if $\mathbf{F}$ is a conservative vector field ($\mathbf{F}(x, y) = \nabla f(x, y)$), then:
  • $\int_C \mathbf{F}(x, y) \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$
  • The line integral depends only on the endpoints of the curve (path-independent)

Surface Integrals and Green's Theorem

Applications of Green's theorem

• Green's theorem relates a line integral around a closed curve $C$ to a double integral over the region $D$ enclosed by $C$
  • $\oint_C P dx + Q dy = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA$
  • $P(x, y)$ and $Q(x, y)$ are functions with continuous partial derivatives in $D$
• Assumptions for Green's theorem:
  1. The curve $C$ is piecewise smooth and simple closed (no self-intersections)
  2. The region $D$ is simply connected (no holes)
• Applications of Green's theorem include:
  • Evaluating line integrals by converting them to double integrals (simplification)
  • Proving vector fields are conservative or path-independent (curl-free)

Calculation of surface integrals

• Surface integrals integrate a function over a surface (flux through a surface)
  • Denoted as $\iint_S f(x, y, z) dS$ where $dS$ represents the surface area element
• Parametrization of a surface represents the surface as a function of two parameters $u$ and $v$
  • $\mathbf{r}(u, v) = (x(u, v), y(u, v), z(u, v))$, where $(u, v)$ is in a domain $D$
• Surface integral with parametrization:
  • $\iint_S f(x, y, z) dS = \iint_D f(\mathbf{r}(u, v)) \left\|\frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}\right\| du dv$
• Flux of a vector field $\mathbf{F}$ through a surface $S$:
  • $\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_S \mathbf{F} \cdot \mathbf{n} dS$ where $\mathbf{n}$ is the unit normal vector to the surface (orientation)
• Work done by a force field $\mathbf{F}$ over a surface $S$:
  • $W = \iint_S \mathbf{F} \cdot d\mathbf{S}$