exm3 last lect

4 thrms:

• stokes thm
• div thm
• greens thm
• Fundemental theron line $\int$

only works for V being conserative

Orientations

1. Choice of tan vect
2. surface: voice of normal vector
3. how apply to 4 thrms?:
   1. greens thrm, counterclockwise
   2. divergence: normal pts outward
   3. stokes, $\sum$ and its partial are related by right hand rule

Loops and Paths

• concat/reverrsal
• conserative vector fields:
  • if conserative
  • check conserative
  • integrate on loops: 0 if V conserative

Dimensions

• ?D object in ?D space
  • 1D 1D ? func
  • 1d 2D ? plane curve
  • 1D 3D ? space curve
  • 2D 2D ? region
  • 2D 3D ? surface
  • 3D 3D ? volume

Integrals

• ${\int }_{C}f(x,y)dS$
• ${\int }_{C}V\cdot ds$
• $\int {\int }_{B}f(x,y)dA$ : if f(x,y)=1 then is area of R
• $\int {\int }_{S}f(x,y,z)dA$ surface int, surface area of R
• $\int {\int }_{e}V\cdot \vec{n}dS$ Flux
• $\int \int {\int }_{V}f(x,y,z)dV$ normal triple int

Cord sys

• Cartesian
• polar: r (jacobian)
• cylindrical: $r
• spherical

Lagrange mult

• Over single curve
• multi curves

Others

• critical points? (panic geq and leq 0 stuff)
• Tangent planes (normal vec from crossing partials)
• Chain rule / total derative (THERE WILL BE PROBLEM ON TOTAL DERATIVE )
• graident + directional deratives
• defition of differentiable integrals?
• deratives along cruves
• limits along curves from 2d limits
• making planes/lines frm info
• making @param w/ PP NP TP