Divergence Theorem Examples

Why This Matters

The Divergence Theorem is one of the most powerful tools in vector calculus, connecting what happens inside a region (a volume integral of divergence) to what happens on its boundary (a flux integral across a closed surface). You're being tested on your ability to recognize when this theorem simplifies calculations, choose appropriate coordinate systems, and set up the corresponding integrals correctly. Understanding how different geometric shapes interact with vector fields reveals the deep relationship between local behavior (divergence at a point) and global behavior (total flux through a surface).

These examples aren't just exercises in computation—they demonstrate core principles like symmetry exploitation, coordinate system selection, and boundary orientation. When you encounter a Divergence Theorem problem on an exam, your first instinct should be to ask: "What symmetry does this region have, and which coordinate system matches it?" Don't just memorize the shapes—know why each one calls for a particular approach and what computational advantages it offers.

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Regions with Cartesian Symmetry

These shapes align naturally with Cartesian coordinates, featuring flat faces and constant integration limits. The key advantage is that bounds don't depend on other variables, making the triple integral straightforward to evaluate.

Solid Cube

• Constant integration limits—all six faces are perpendicular to coordinate axes, so $\int_0^a \int_0^a \int_0^a (\nabla \cdot \mathbf{F}) \, dV$ requires no variable substitution
• Direct surface integral comparison—each face has an outward normal parallel to a coordinate axis ($\pm \hat{i}, \pm \hat{j}, \pm \hat{k}$), making flux calculations on individual faces simple
• Foundational example for verifying the Divergence Theorem by computing both sides independently and confirming equality

Rectangular Prism

• Generalized Cartesian region—extends the cube concept with dimensions $a \times b \times c$, maintaining constant limits $\int_0^a \int_0^b \int_0^c$
• Separable integrals—when $\nabla \cdot \mathbf{F}$ factors into functions of single variables, the volume integral becomes a product of three one-dimensional integrals
• Standard test case for building intuition before tackling curved surfaces or variable bounds

Tetrahedron

• Triangular faces with linear bounds—integration limits become functions of other variables (e.g., $z$ from $0$ to $1 - x - y$), requiring careful setup
• Compact visualization—four vertices and four faces make it manageable to verify flux through each triangular surface
• Tests mastery of variable limits in Cartesian coordinates without introducing curved boundaries

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Regions with Spherical Symmetry

Spherical coordinates $(r, \theta, \phi)$ transform these regions into simple bounds. The volume element $dV = r^2 \sin\phi \, dr \, d\theta \, d\phi$ and radial symmetry often collapse complex vector fields into manageable expressions.

Spherical Shell

• Concentric sphere boundaries—the region $a \leq r \leq b$ has constant angular limits ($0 \leq \theta \leq 2\pi$, $0 \leq \phi \leq \pi$), isolating all complexity in the radial direction
• Radial vector fields simplify dramatically—for $\mathbf{F} = f(r)\hat{r}$, the divergence $\nabla \cdot \mathbf{F} = \frac{1}{r^2}\frac{d}{dr}(r^2 f)$ exploits full spherical symmetry
• Classic inverse-square law application—demonstrates why $\mathbf{F} = \frac{\hat{r}}{r^2}$ has zero divergence everywhere except the origin

Hemisphere

• Half-sphere with planar cap—bounded by $r \leq R$ and $0 \leq \phi \leq \frac{\pi}{2}$, requiring attention to the flat circular base as part of the closed surface
• Symmetry reduces angular integration—many integrands vanish or simplify when integrated over the hemisphere due to odd-function cancellation
• Common exam setup where you must include flux through the base disk to form a closed surface

Ellipsoid

• Stretched spherical symmetry—defined by $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \leq 1$, requiring either a coordinate transformation or careful Cartesian parameterization
• Jacobian scaling—transforming to $u = x/a$, $v = y/b$, $w = z/c$ maps the ellipsoid to a unit sphere with $dV = abc \, du \, dv \, dw$
• Tests coordinate transformation skills and understanding of how divergence scales under non-uniform stretching

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Regions with Cylindrical Symmetry

Cylindrical coordinates $(r, \theta, z)$ are ideal when a region has rotational symmetry about one axis. The volume element $dV = r \, dr \, d\theta \, dz$ handles circular cross-sections efficiently.

Cylindrical Shell

• Annular cross-section—bounded by $a \leq r \leq b$, $0 \leq \theta \leq 2\pi$, and $0 \leq z \leq h$, with all limits constant
• Rotational symmetry—vector fields of the form $\mathbf{F}(r, z)$ (independent of $\theta$) yield $\theta$-integrals that factor out as $2\pi$
• Divergence formula—$\nabla \cdot \mathbf{F} = \frac{1}{r}\frac{\partial}{\partial r}(rF_r) + \frac{1}{r}\frac{\partial F_\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}$ must be memorized for these problems

Cone

• Variable radius with height—the region $0 \leq r \leq \frac{R}{h}z$, $0 \leq z \leq h$ has $r$-bounds that depend on $z$
• Singular apex—the point at $z = 0$ (or the tip) requires careful treatment; divergence behavior near this point can reveal important physics
• Non-uniform cross-section tests your ability to handle $r$-limits that vary, distinguishing it from the cylindrical shell

Paraboloid

• Curved boundary in $r$-$z$ plane—typically $z = r^2$ or $r^2 = az$, creating a region where $r$-limits depend on $z$ (or vice versa)
• Gravitational and potential applications—paraboloidal surfaces appear in physics problems involving reflectors and equipotential surfaces
• Orientation matters—whether the paraboloid opens upward or downward changes the integration setup and outward normal direction

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Regions with Complex Topology

These shapes challenge standard approaches due to non-convexity, holes, or unusual parameterizations. The Divergence Theorem still applies, but careful attention to surface orientation and closed boundaries is essential.

Torus

• Doughnut topology—the "hole" means the surface has genus 1, requiring parameterization by two angles (poloidal and toroidal)
• Non-convex region—outward normals point in different directions on inner versus outer portions of the surface
• Toroidal coordinates—using $(R + r\cos\phi)\cos\theta$, $(R + r\cos\phi)\sin\theta$, $r\sin\phi$ where $R$ is major radius and $r$ is minor radius

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Quick Reference Table

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Self-Check Questions

1. Which two regions both use cylindrical coordinates but differ in whether the $r$-bounds are constant or variable? What does this difference mean for setting up the integral?

2. For a radial vector field $\mathbf{F} = r^n \hat{r}$, which region would most naturally exploit the symmetry to simplify the divergence calculation? Why?

3. Compare and contrast the Tetrahedron and the Solid Cube: both use Cartesian coordinates, so what makes one more computationally challenging than the other?

4. If an FRQ asks you to verify the Divergence Theorem by computing both the volume integral and the surface integral independently, which region would you choose for the cleanest calculation? Justify your choice.

5. The Hemisphere requires including a flat circular base to form a closed surface. Why is this necessary for applying the Divergence Theorem, and how would you compute the flux through this base?