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University Math Lab: Systems & Structures FREE
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Divergence & Divergence Theorem
Learning Objectives
- Compute divergence of vector fields
- Apply Divergence Theorem
- Interpret physically
Key Concepts
Divergence
div F⃗ = ∇·F⃗ = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
Divergence Theorem
∫∫∫ ∇·F⃗ dV = ∫∫ F⃗·n̂ dS
Theory
**Divergence:** ∇·F⃗ = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z
**Interpretation:** Measures the "expansion" of a vector field at a point.
- div > 0: Source (field expands)
- div < 0: Sink (field contracts)
- div = 0: Incompressible
**Divergence Theorem:** ∮∮_S F⃗·dS⃗ = ∫∫∫_V ∇·F⃗ dV
Converts surface integral to volume integral.
Examples
div of F⃗ = ⟨x², xy, z⟩
Solution: ∂(x²)/∂x + ∂(xy)/∂y + ∂(z)/∂z = 2x + x + 1 = 3x + 1