Circulation flux divergence -- caluclus with curves

Key ideas:

  • Curl is infinetestimal circulation just as divergence is infetestimal flux.

Fundamentals / building blocks:

  • Parametrize curves:

    • Parametrization:
      a map $\gamma $ so that curve $C$ is image of $\gamma$ on $[a,b]$ and $\gamma$ is continuous on $[a,b]$.
    • regular parametrization:
      same but $\gamma$ must be $C^1$ and $\gamma^\prime(t)\neq 0$ in the open interval $(a,b)$.
    • simple regular:
      add to regular the constraint that $\gamma$ is injective except when $\gamma(a)=\gamma(b)$1.
    If $\gamma(a)=\gamma(b)$ then we say $\gamma$ is closed.2

  • Line integrals


  • Vector fields:


Circulation


Flux


Green’s theorem



  1. 1.so for every $t$ there is a point mapped to on the curve.
  2. 2.think how the curve is closed when the tips touch.
Author

ChiatzenW

Posted on

2022-03-15

Updated on

2022-03-15

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