So, in both cases, we need the vector field to be defined not only, I mean, the left hand side makes sense if a vector field is just defined on the curve because it's just a line integral on c.

了解这两种表述后，我们不仅需要向量场，就是左边这里，这是曲线c上的线积分，向量场在曲线上有定义。

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Let's say that we had the same c, but now the vector field.

假设c还是一样的，但现在的向量场变为。

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At the forefront of the pioneers in the field was the Greek mathematician and philosopher Claudius Ptolemaeus (c. AD 90 ?168), more popularly known to history as Ptolemy.

这一领域的先驱中最早的是希腊数学家和哲学家克劳迪•托勒密(约公元90年～168年)，历史上多称他托勒密。

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C. congratulated each other on the field.

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