MATH 32A Lecture Notes - Lecture 6: Ellipse, Dot Product, Product Rule
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Lecture 6: arc length of a curve r(t) = = position of some object at time t. We say that the limit as t t o of r(t) is the vector l if the limit as t t o of r(t) - l = 0 (which has 0 magnitude) so we can also write limit as t t o of ||r(t) - l|| = 0. Theorem: suppose r(t) = . Limits are computed as one component at a time. Then limit as t 0 of r(t) = <1, 0, 1>. In other words, to compute derivatives of vvfs, just do it component-wise. d/dt[r 1 (t) + r 2 (t)] = dr 1 /dt + dr 2 /dt. d/dt(cr(t)) = cdr/dt for a constant scalar c. d/dt[r(g(t))] = g"(t)r"(g(t)). g"(t) is a scalar, r"(g(t)) is a vector. Scalar product rule: d/dt[f(t)r(t)] = f"(t)r(t) + f(t)r"(t).
