1873_solutions

Is an increasing function and if h is a closed

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Unformatted text preview: T t  T t  0. In this event, the curvature K t of the curve  at the number t is defined by the equation Tt . Kt  t In the event that K t 0 then the principal normal N t of  at the number t is defined by the equation Tt Tt Nt   . Tt Kt  t b. Prove that if  t exists and K t 0 then  t  tTt  t 2 KtNt Exercises on Integrals on Curves 1. Point at each of the equations  1 t  t cos t, t sin t, t and  2 t  t 2 cos t sin t, t 2 sin t, t 2 cos t and f x, y, z  xy, yz, zx and supply them as definitions to Scientific Notebook. Taking the domain of each of the curves  1 and  2 to be the interval 0, 1 , use Scientific Notebook to evaluate each of the intervals Þ f x, y, z  d x, y, z and 1 Þ f x, y, z  d x, y, z . For example, the first of these integrals is 2 Þ 0 f t cos t, t sin t, t 1   t dt. Are the two integrals you have just evaluated equal to one another? 2. For each of the following curves  and point a, b in R 2 , evaluate the integral 1 2 Þ 1 y b ,x x, y a, b 2 Ask Scientific Notebook to show you a sketch of each curve. a. We define 405 a  d x, y .  t  3 cos t, 3 sin t for 0 t 2 and a, b  1, 1 . b. We define for 0 t t  ,0 . 1  2 cos t cos t, 1  2 cos t sin t t  2 and a, b  2, 0 . 1  2 cos t cos t, 1  2 cos t sin t t  2 and a, b  4, 0 . 1  2 cos t cos t, 1  2 cos t sin t 2 and a, b  1 2 c. We define for 0 t d. We define for 0 t 3. Given that  t   1 t,  2 t , ,  n t for each t in a given domain a, b and given that each of the functions  j is Riemann integrable on a, b we define Þ a  t dt  Þ a  1 t dt, Þ a  2 t dt, , Þ a  n t dt b b b b . a. Prove that the integral just defined has the properties of linearity and additivity that we obtained for Riemann integrals of real functions. These properties follow at once from the definitions. b. State and prove an analog of the fundamental theorem of calculus for this kind of integral. Suppose that t  1 t , 2 t , , n t for a t b and that each function j has a Riemann integrable derivative on a, b . Then Þa b t dt  Þa b t, 1  Þa  1  b b 1 2 t , , t dt, Þ b 2 a, 1 t dt t dt, , Þ b a n b 2 2 b n a t dt a , , n b n a a c. Assuming that Þ a  j t dt  q j b for each j, obtain the identity 2 Þ a  t dt b  n Þa b qjj t dt. j1 We observe that Þ a  t dt b 2  Þ a  1 t dt b n  n q2  j j1 j1 406 2 Þ a  2 t dt b  q j Þ  j t dt  b a 2 Þa b  Þ a  n t dt b n qjj t j1 dt. 2 d. By applying the Cauchy-Schwarz inequality to the latter expression, deduce that Þ a  t dt Þa b b t dt. We observe that Þ a  t dt b 2   n Þa b dt j1 Þ a  t dt Þ a b Þa n b qjj t b t n  2 t dt j q2 j j1 j1 dt. and it follows that Þ a  t dt b Þa b t dt. Some Exercises on Exact Functions 1. a. Find a potential function for the function f defined to be 2xz , 2yz  sin yz  yz cos yz, log x 2  y 2  y 2 cos yz  2z 2  y2 x2  y2 x at every point x, y, z of R 3 for which x 2  y 2 0. We are looking for a function F for wh...
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