Lecture 3 23 (x, y) = (t2, t3) (x, y) = (t3, t3) Figure 1.17. Two curves with a vanishing tangent vector at t = 0. where t lies in the interval [a, b], whose endpoints a and b may be ±∞. In this representation, the circle discussed above would be written as (x, y, z) = (cos t, sin t, 0) with 0 ≤ t ≤ 2π. If we replace the equation z = 0 with z = t, we obtain not a circle, but a helix it takes a little more imagination to picture this as the intersection of two surfaces. We could also multiply the expressions for x and y by t to describe a spiral on the cone, whose implicit representation is again not immediate. Exercise 1.12. Find two equations whose common solution set is the helix. If we expect our curve to be smooth, we must impose certain conditions on the coordinate functions fi. The first condition is that each fi be continuously differentiable this will guarantee the existence of a continuously varying tangent vector at every point along the curve. However, if we do not impose the further requirement that this tangent vector be non-vanishing, that is, that (f 1 )2 +(f 2 )2 +(f 3 )2 = 0 holds everywhere on the curve, then the curve may still fail to be smooth. As a simple but important example of what may happen when this condition is violated, consider the curve (x, y) = (t2, t3). The tangent vector (2t, 3t2) vanishes at t = 0, which appears as a cusp at the origin in Figure 1.17. So in this case, even though f1 and f2 are perfectly smooth functions, the curve itself is not smooth.

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