Abstract
In this chapter, we develop the mathematical tools needed to model and study a moving object. The object might be moving in the plane:
Spiderweb segments dangle in the shape of catenary curves, exemplifying aspects of the general theory of curves presented in this chapter.
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Notes
- 1.
Many calculus books enclose vectors in pointed brackets, like \(\left \langle x(t),y(t),z(t)\right \rangle\), but we will always use round parentheses. The term “vector” is a synonym for “element of Euclidean space.” In some situations, it is best visualized as a point, and in others as an arrow with its tail at some particular position. Names of vectors (and vector-valued functions) will be typeset in boldface throughout Chaps. 1 and 2. For handwritten math, we recommend over-arrows rather than bold, like \({\boldsymbol \gamma }\).
- 2.
Many calculus books denote this by x ⋅ y and call it the dot product.
- 3.
This proof uses the relative definitions of “neighborhood,” “open,” and “closed” explained in Sect. 1 of the appendix.
- 4.
Some authors define torsion as the negative of this definition.
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© 2016 Springer International Publishing Switzerland
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Tapp, K. (2016). Curves. In: Differential Geometry of Curves and Surfaces. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-39799-3_1
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DOI: https://doi.org/10.1007/978-3-319-39799-3_1
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