Divergence Theorem and Applications

Introduction

• Recap of Green's theorem from section 16.5.
• Two versions of Green's theorem discussed:
  • One generalizes to Stokes' theorem.
  • The other generalizes to the Divergence theorem.

Divergence Theorem

• The Divergence theorem relates the flow of a vector field out of a closed surface to the behavior of the field inside the surface.
  • Notation:
  • Let 'D' be a positively oriented boundary of a planar region 'D'.
  • Generalization to a solid region in $R^3$, denoted as 'E', with boundary 's'.
• Surface integral expression: $ext{flux} = ext{Surface Integral} ext{ F} \cdot ext{n} ext{ ds}$
  • Where 'n' is the normal vector to the surface.
• Divergence theorem expressed mathematically as: $\int<em>{s} ext{F} \cdot ext{n} ext{ds} = \int</em>{E} ext{div F} ext{dV}$
  • Here, 's' is the boundary of solid region 'E'.

Generalization Details

• The solid region 'E' can be of types:
  • Type 1: 'z' between functions of 'x' and 'y'.
  • Type 2: 'x' between functions of 'y', with 'z' as functions of 'y' and 'z'.
  • Type 3: 'y' trapped between functions of 'x' and 'z'.
  • Simultaneously type 1 and type 3, with conditions for types 2 and 3, forms a simple solid region.
• Examples of Simple Solid Regions:
  • Sphere, box, ellipsoid, etc.
• Orientation requirements:
  • For closed surfaces, the unit normal vector must point outward.

Formal Statement of the Divergence Theorem

• Let 'E' be a solid region with boundary 's' (positively oriented).
• Let 'F' be a vector field defined on an open region that contains 'E'.
• The formal expression of the divergence theorem: $\int<em>{s} ext{F} \cdot ext{n} \text{ds} = \int</em>{E} ext{div F} \text{dV}$
  • Div 'F' must be well-defined for all points in 'E'.

Example 1: Divergence Calculation

• Identify vector field $ext{F} = (x, y, x^2)$ on unit sphere defined by $x^2 + y^2 + z^2 = 1$.
• Surface integral becomes:
  $\int<em>{s} ext{F} \cdot ext{ds} = \int</em>{B} ext{div F} \text{dV}$ where B is the unit ball.
• Calculate divergence:
  • $ext{div F} = \frac{\partial}{\partial x}(x) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(x^2) = 1$.
• Integral simplifies: $\int_{B} 1 \text{dV}$ results in the volume of the unit ball.
  • Volume formula: $V = \frac{4}{3} \pi r^3$, therefore, for radius 1: $V = \frac{4}{3} \pi$.

Example 2: Surface and Region Description

• Vector field $ext{F} = x y extbf{i} + \left(y^2 + e^{x} z^2\right) extbf{j} + \sin(x y) extbf{k}$.
• Boundary 's' described by parabolic cylinder $z = 1 - x^2$ and planes $z = 0, y = 0, y + z = 2$.
• Surface:
  • Visualized in coordinate planes: parabolic cylinder bounded appropriately by specified planes.
  • Encounter four sides of varying forms (bottom, parabola, two planes).

Iterated Integral Setup

• Set bounds:
  • Buckets for 'x': between -1 and 1.
  • For 'z': functions of 'x' with ranges $[0, 1 - x^2]$.
  • For 'y': between $[0, 2 - z]$.
• Integral set as: $\int<em>{-1}^{1} \int</em>{0}^{2 - z} 3y \text{d}y \text{d}z \text{d}x$
  • Allows integration to be performed iteratively.

Generalization of Divergence Theorem for Nested Surfaces

• Consider two surfaces ‘S1’ (inner surface) and ‘S2’ (outer surface) with corresponding normals ‘n1’ and ‘n2’.
• The divergence theorem applied as:
  $\int<em>S ext{F} \cdot ext{n} ext{ds} = \int</em>S (- ext{F}\cdot ext{n}<em>1 + ext{F}\cdot ext{n}</em>2) ext{ds}$.

Electric Flux Example

• Electric field $ext{E} = \frac{Q}{4 \pi \epsilon \text{R}^2}$ at distance from charge at origin.
• Applying the divergence theorem with a sphere of radius small 'a' cut out.
• Usage involves ensuring no discontinuities exist.
• Formation concludes with expression:
  $\Phi<em>E = \oint</em>{s} ext{E} \cdot ext{dS} = \frac{Q}{\epsilon}$.

Homework Problems Discussion

• Discuss integrating function: $\int_C (y + 5 \sin{x}) ext{dx} + (z^2 + 5\cos{y}) ext{dy} + (x^3) ext{dz}$
  • With curve 'C' defined as $R(t) = (\sin{t}, \cos{t}, 2 \sin{t} \cos{t})$.
• Exploration of surface orientation and projection of curve. Coordinates around (0,0) interval noted.
• Discuss behavior of projection, orientation and necessary traversals of curves for proper integration setup.

Conclusion & Review

• Emphasize importance of understanding divergence theorem and its applications, providing meaningful examples to solidify comprehension.
• Ensure readiness for integration techniques discussed previously as problems approach him.