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$\mathbf x$ describes a parametrized $C^r$ curve.

If $f: \overline{I} \rightarrow I, \overline{t} \mapsto t$ is a valid (I don't know the exact translation, in german it's zulässig) parameter transformation, then $\overline{\mathbf x}$ describes the same curve as $\mathbf x$

$\overline{\textbf x}: \overline{\mathbf I} \rightarrow \mathbb R^3 \\ \overline{t} \mapsto {\textbf x}(f(\overline{t}))=\overline{\mathbf x}(\overline{t})$

because of

$\frac{d\overline{\mathbf x}}{d\overline{t}}=\frac{d \mathbf x}{dt}|_{f(t)} \cdot \frac{df}{d\overline{t}}|_{\overline{t}} \not= \mathbf o $.

What does $|_{f(t)}$ and $|_{\overline{t}}$ mean?

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It means: "Evaluate the function at the point …."

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