Let y: 1 + R2 be a regular parametrised curve which we write as y(t) =...
79. Parametrised Surface 1. Consider the parametrised surface defined by: x 2u, y u2+ v (a) Find a vector normal to the surface in terms of u and v. (b) For what values of u and v is the surface smooth? (c) Find the equation of the tangent plane to the surface at (0,1, 1) 79. Parametrised Surface 1. Consider the parametrised surface defined by: x 2u, y u2+ v (a) Find a vector normal to the surface in terms...
l maps is a quotient map. 4, Let ( X,T ) be a topological space, let Y be a nonempty set, let f be a function that maps X onto Y, let U be the quotient topology on induced by f, and let (Z, V) be a topological space. Prove that a function g:Y Z is continuous if and only if go f XZ is continuous. l maps is a quotient map. 4, Let ( X,T ) be a topological...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto B. (0,1) (1;1) (-1,1) (0,1) v=1 1/2 y-axis y=x2 v-axis v -u b ets t) ide ods.a notteog (0,0) X-axis u-axis (0,0) (e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto...
Let V = P1(R) and W = R2. Let B = (1,x) and y=((1,0), (0, 1)) be the standard ordered bases for V and W respectively. Define a linear map T:V + W by T(P(x)) = (p(0) – 2p(1), p(0) + p'(0)). (a) Let FEW* be defined by f(a,b) = a – 26. Compute T*(f). (b) Compute [T]y,ß and (T*]*,y* using the definition of the matrix of a linear transformation.
6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0<x<a, 0<t (2') u(0,y, t)-gi(v), u(a,y,t)-89(v) 0 <y<b, o<t (3) Show that the steady-state solution involves the potential equation, and indicate how to solve it. 6. Suppose that, instead of boundary conditions Eqs. (2) and (3), we have u(x, o, t) -f^(r), u(r, b, t)() 0
5. Let (S2,F,P) be a probability space and let {W(t),t 2 0) be Brownian mo- tion with respect to the filtration Ft, t 2 0. By considering the geometric Brownian motion where Q R, σ > 0, S(0) > 0. Show that for any Borel-measurable function f(y), and for any 0 〈 8くthe function 2 2 g(x) =| f(y) da 0 satisfies Ef(S(t))F (s)-g(S(s)), and hence S(t) has a Markov property. We may write qlx as q We may write...
Question 4 (Geodesics on surfaces of revolution) Let S be a surface of revolution and consider for it the parametrization x(u, v) ((v) cos u, p(v) sin u, ^(v) Assume in addition that (a)2 +()21 (a) Prove that a curve a(t) = x(u(t), v(t)) is a geodesic of S if and only if it satisfies dip 1 ü2 dv p dip p(u)2 0, dv where here and in what follows the dot denotes derivative with respect to t 5 marks...
a. A function f: A B is called injective or one-to-one if whenever f (x) f(u) for some z, y A then y. Which of the following functions are injective? In r-y. That is Vr,y E A f()-f(u) each case explain why or why not i. f:Z Z given by f(z) 3 7 ii. f which maps a QUT student number to the last name of the student with that student number. b. Suppose that we have some finite set...
() At)x()B(f)u() Consider the following time-varying system y(t) C(f)x(t) where x) R", u(t)E R R 1 1) Derive the state transition matrix D(t,r) when A(f) = 0 0 sint 2) Assume that x(to) = x0 is given and u(f) is known in the interval [to, 4] Based on these assumptions, derive the complete solution by using the state transition matrix D(f, r). Also show that the solution is unique in the interval [to, 4]. 3) Let x(1) 0 and u(f)...