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.↩