... points¹

These are the common zeros of the pair $P(x, y)$ and $Q(x, y)$.

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... equation²

This means a trajectory (solution curve) starting at a point of $\Omega$ stays in $\Omega$ for all times.

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... by³

To avoid introducing more notation we are being notationally imprecise. The expression $\gamma_1(t)\cdot\gamma_2(t)$ inside the brackets refers to the map $t\mapsto \gamma_1(t)\cdot\gamma_2(t)$.

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... covering⁴

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... path⁵

The path would have to satisfy some regularity condition such as being piecewise continuously differentiable. However since this is merely supposed to be a motivation we shall brush aside these technicalities.

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... calculus⁶

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... Dieck⁷

See also R. Stöcher and H. Zeischang, Algebraische Topologie, B. G. Teubner, Stuttgart (1988) 306-325.

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