Lecture 5 and 6 Vector calculus Integration II

Lecture 5 and 6: Vector calculus: Integration II 1. Surfaces (Sec. 10. 5) • Parametric representation of a surface • Tangent plane and normal vector to a surface 2. Surface integral (Sec. 10. 6) • Example: Flux through a surface • Orientation of a surface • One more example: The surface-area of a sphere 3. Triple integrals (Sec. 10. 7): One example tonight 4. Divergence theorem (Gauss theorem) (Sec. 10. 8) 5. Applications of the divergence theorem: (Sec. 10. 9) • Flux out of a volume • Flux/fluid flow • Continuity equation • Diffusion equation • Harmonic functions and Laplace equation 6. Stokes’ theorem (Sec. 10)

Surfaces: Orientation

Non-orientable surfaces: The Möbius strip and the Klein bottle The Möbius strip The Klein bottle

Surface integrals, an example: Surface-area of a sphere

Divergence theorem

Divergence theorem: Geometric ‘intuitive’ interpretation

Divergence theorem: Proof, the LHS

Divergence theorem: Proof, the RHS

Divergence theorem: Verification

Divergence theorem: Verification

Divergence theorem: Verification

Some more useful trigonometric integrals…

Applications of the divergence theorem: Fluid flow

Applications of the divergence theorem: Continuity equation

Applications of the divergence theorem: Diffusion (heat) equation κ = diffusion constant T = temperature (in degrees K) ρ(r) = density (gr/cm 3) c = heat capacity (JK-1 gr-1)

Applications of the divergence theorem: Diffusion (heat) equation

Applications of the divergence theorem: Diffusion (heat) equation

Applications of the divergence theorem: Harmonic functions

Harmonic functions and Gauss’ law

Applications of the divergence theorem: Green’s theorem

Stokes’ theorem

Verification of Stokes’ theorem b

Verification of Stokes’ theorem

Duplicate slides

Harmonic functions

Applications of the divergence theorem: Diffusion (heat) equation