2.7 Determine which of the following are linear operators, and which are not. (i) The parity operator II[f(x)] = f(x). (ii) The translation operator T[f(x)] = f(x+1). (iii) The operator L[f(x)] = f(x) + 1.
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- You are designing an RPG (role-playing game) for a gaming console and have decided to use an open world design, where players can explore the terrain freely, encountering enemies by chance. 8. Your design team has coded this in-game world to exist on the circle x² + y? < 900 Page 3 on the xy-plane. At any point (x, y) in this world you've also associated a danger function d(x, y) that measures how likely it is to encounter an enemy at that point. Thus high values of d(x,y) correspond to dangerous points, while low values of d(x, y) correspond to safe points. If d(x, y) = e¬a²y, find the safest point(s) and most dangerous point(s) in-game.The circumference C of a circle is a function of its radius by C(r) = 2xr. Express the radius of a circle as a function of its circumference. Call this function r(C). r(C) = Preview Find r(187). r(187) = Interpret the meaning: O When the radius is 187, the circumference is r(187) O When the circumference is 187, the radius is r(187)42 Let Ta denote the translation operator (displacement vector d); let D(n,p) denote the rotation operator ( and are the axis and angle of rotation, respectively); and let denote the parity operator. Which, if any, of the following pairs commute? Why? (a) Ta and T (d and d' in different directions). (b) D(n,p) and D(n, ') (ñ and f' in different directions). (c) T₁ and . (d) D(n,) and .
- 7Find the divergence and curl of each of the following vector functions: 1. S = < xy, 2yz, 3zx > 2. E = < y^2, (2xy+z^2), 2yz >Construct the 3D rotation ma- trices for rotations about the x-axis, y-axis, and z-axis by an angle o as described in I section 1.1.5. You should end up with three 3x3 matrices, one for each rotation axis. Demonstrate that your matrices do what they are supposed to do by considering >= 90 deg and applying them to an arbitrary vector A. Draw pictures to illustrate your reasoning.