User:Egm6321.f10.team4.Yoon/Mtg8

 Mtg 8: Thu, 9 Sep 10

[[media: 2010_09_09_13_54_49.djvu | Page 8-1]] Note: [[media: 2010_09_02_14_58_46.djvu | p.6-1]] Linearly independent 1) Vectors(in $$\displaystyle \mathbb R^2 $$ for simplicity) 2) Functions: [[media: 2010_09_02_14_58_46.djvu | p.6-1]]

Read: King et al. Appendix 5 p.511-516 for particular case of Lecture.

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Example: [[media: 2010_09_07_14_59_22.djvu | p.7-3]] continued:

First condition of Exactness: For $$\displaystyle F(x,y,y')=0$$ to be exact, $$\displaystyle F(\cdot) $$ must be in the form:

Example [[media: 2010_09_07_14_59_22.djvu | p.7-2]]:

[[media: 2010_09_09_13_54_49.djvu | Page 8-3]]

Satisfied 1st Condition of exactness (Not necessarily exact yet-Need to satisfy 2nd condition)

Eqs.(2)&(3) [[media: 2010_09_07_14_59_22.djvu | p.7-2]] also satisfy 1st condition of exactness

Eqs.(4)[[media: 2010_09_07_14_59_22.djvu | p.7-2]]:

Example:

Above equation does not satisfy 1st condition of exactness.

(1)&(2)[[media: 2010_09_09_13_54_49.djvu | p.8-2]]: If $$\displaystyle \phi $$exists, what would be the relationship between the two equations?

Can M(x,y) and N(x,y) be arbitrary? No!

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Since

Assuming $$\displaystyle \phi(\cdot, \cdot)$$ is smooth

Second condition of Exactness: