Divergence Theorems and Green's Theorem

Green's Theorem

  • Definition: A fundamental theorem relating a surface integral over a region to a line integral over its positively oriented boundary.
    • Mathematical Representation: $\iintS \mathbf{F} \cdot d\mathbf{A} = \ointC \mathbf{F} \cdot d\mathbf{r}$.
    • Where:
    • $S$ is the surface bounded by the curve $C$.
    • $ ext{F}$ is a vector field.
    • $d\mathbf{A}$ is the area element on $S$.
    • $d\mathbf{r}$ is the differential line element along $C$.

Generalization of Green's Theorem to $\mathbb{R}^3$

  • Statement: The surface integral over a solid region can be expressed as a volume integral of the divergence over that region.
    • Mathematical Representation: $\iintS \mathbf{F} \cdot d\mathbf{A} = \iiintE \text{div } \mathbf{F} \, dv$.
    • Where:
    • $E$ is the solid region.
    • $S$ is the boundary surface of region $E$ with positive outward orientation.

Divergence Theorem

  • Definition: A theorem that relates the flow of a vector field through a closed surface to the behavior of the vector field in the volume enclosed by the surface.
    • Mathematical Representation: $\iintS \mathbf{F} \cdot d\mathbf{A} = \iiintE \text{div } \mathbf{F} \, dv$.
    • Let $E$ be a simple solid region, and $S$ be the bounding surface of $E$ with positive outward orientation.
    • $ ext{F}$ must be a vector field defined with components that have continuous partial derivatives over an open region.

Example Problem

  • Find the flux of the vector field $\mathbf{F}(x, y, z) = z \mathbf{i} - x \mathbf{j} + y \mathbf{k}$ over the unit sphere defined by the equation $x^2 + y^2 + z^2 = 1$.
    • The flux can be calculated by:
    • $\Phi = \iintS \mathbf{F} \cdot d\mathbf{A} = \iintS d\mathbf{r} \cdot \mathbf{F}$.
    • By applying the divergence theorem, it can also be found as: $\Phi = \iiint_E \text{div } \mathbf{F} \, dv$.

Additional Example

  • Better understanding through the divergence equation via the example with specific components $\text{div } \mathbf{F}(x, y, z) = \frac{\partial F1}{\partial x} + \frac{\partial F2}{\partial y} + \frac{\partial F_3}{\partial z}$.

Types of Surfaces

  • Discuss the parabolic cylinder as bounding surfaces for the region $D$.
    • Equation: $z = 1 - x^2$ and plane $x + y + z = 2$.
    • Examples of coordinates for specific regions are given within the surface integrations.
    • Setup iterated integrals for volume calculations over defined surfaces: $dV = dx \, dy \, dz$.

Homework Problems

Problem 1

  • Given vector field: $\mathbf{F}(x, y, z) = 4x \mathbf{i} + xy \mathbf{j} + 5xz \mathbf{k}$.
    • Determine the flux through the bounding surfaces of a cube situated at the origin with dimensions defined in space.
    • Use the relation of volume flux and divergent displacement: flux $= \iiint_E \text{div} \mathbf{F} \, dv$.

Problem 2

  • Expand on surface integral computations involving various surfaces $S1$, $S2$, etc., defined with certain limits.

Divergence in Fluid Flow

  • Consideration of divergence $\text{div } \mathbf{F}$ in the context of fluid dynamics at a point $P_o$ with associated flow implications:
    • If $\text{div} \mathbf{F}(Po) > 0$, fluid flows outward at $Po$.
    • If $\text{div} \mathbf{F}(P_o) < 0$, fluid is drawn inward (sink).

Mathematical Definitions

  • Vector field representation: $\mathbf{F} = (f1(x,y,z), f2(x,y,z), f_3(x,y,z))$.
  • Components must meet continuity and differentiability requirements within the region of integration, meeting prerequisites for the application of divergence and theorems regarding flux.

Closing Notes

  • Importance of establishing orientation, calculating various integrals for flux, and understanding the implications on flow dynamics in physical environments, underscoring the dynamic interaction between vector fields and geometric regions.