Math252CalculusIII:VectorCalculusOverview by: javier

Math 252 Calculus III: Vector Calculus Overview

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by: javier

Line Integrals common forms and meaning

common forms and meaning

∫ ⃗F · d⃗r C

common forms and meaning

∫ ⃗F · d⃗r ∫

C

⃗F · ⟨dx, dy, dz⟩ C

common forms and meaning

∫ ⃗F · d⃗r C

∫ ⃗F · ⟨dx, dy, dz⟩ C

∫ M dx + N dy + P dz C

common forms and meaning ∫ ⃗F · d⃗r C

∫ ⃗F · ⟨dx, dy, dz⟩ C

∫ M dx + N dy + P dz C

I M dx + N dy + P dz C

often represent quantities such as WORK ex. work by wind on a bird flying

Line Integrals common techniques for computing

common techniques for computing ∫

∫ ⃗F · d⃗r =

M dx + N dy + P dz C

C

□ Parametrize path and change integral to t’s □ (if curl(⃗F) = 0) find the potential for the field. Evaluate potential at path endpoints... □ (if path C is closed and in 2D) use greens theorem ∫ ∫ M dx + N dy = (Nx − My ) dA C

R

□ (if path C is closed and in 3D) use stokes theorem ∫ ∫ M dx + N dy + P dz = curl(⃗F) · ⃗n dS C

S

Surface Integrals common forms and meaning

common forms and meaning

∫ ⃗F · d⃗S S

∫ ⃗F · ⃗n dS S

if S is given by z = g(x, y) ∫ ⃗F · ⟨gx , gy , −1⟩ dA

± S

often represent quantities such as wind, rain, electromagnetic fields, ex. how much wind goes trough a surface

Surface Integrals common techniques for computing

common techniques for computing

∫ ⃗F · ⃗n dS S

∫ □ use ± S ⃗F · ⟨gx , gy , −1⟩ dA □ (if S is closed) use Gauss’s Divergence Theorem ∫ ∫∫∫ ⃗F · ⃗n dS = div(⃗F)dV S

E

□ (if S is almost closed) add the surface that completes S into a closed surface use Gauss’s Divergence

Theorem the compute the surface integral over the added piece to subtract from flux over closed surface. □ (if⃗F is the curl of some function⃗G, ie if⃗F = curl(⃗G)) find the (path) boundary for S and use stokes theorem, computing the path integral of⃗G over the boundary rather than the flux of⃗F over surface S (duh!) ∫

∫

S

I curl(⃗G) · ⃗n dS =

⃗F · ⃗n dS = S

⃗G · d⃗r C

Good to Remember a few useful ideas

a few useful ideas □ if z = g(x, y) is some surface S then √ dS =

(gx )2 + (gy )2 + 1 dA

() □ if curl ⃗F = 0 then ⃗F is a gradient field □ (converse..)

( ( )) curl ∇ ⃗F = 0

□ also..

( ( )) div curl ⃗F = 0