Problem 9) Y u may/should use mathematica or alpha to help taking derivatives Bipolar co-ordinates are useful for describing various engineering problems (antenna/ microphone design, heat sources near a circular boundary, two parallel cylindrical thermal or electrical conductors, etc) The transformation from Bipolar co-ordinates (t,s) to Cartesian co-ordinates (x,y) is described by x-a(sinh[t])/(cosh[t]-cos[s]) y-a(sin[s)/(cosh[t-cos[s]) For purposes of this problem, set a-1 Curves of constant t and s describe "Apollonian circles" (colored red and blue) and the corresponding x.y value is the intersection of the pair of Apollonian circles for each t,s. In the figure (shamelessly stolen from wikipedia), the red circles (constant s) each pass through the two points (-1,0) and (1,0) are centered at (x.y)-(0,cot(s)) and have radius 1/sin2(s). The blue circles (constant t), are centered at (x.y)-(coth(t),0) and have radius 1/sinh(t) i) Consider the point (x.y) (1/2,0). First show the values (t,s) (1/2 Log[9], Tt) satisfy ii) Next, show that the hypotheses of the implicit function theorem are satisfied so that the co-ordinate transformation for x,y=1/20 ("Log" is natural log) we are sure to have a unique inverse function (t[(x,y)].s[(x,y)]) that satisfies the coordinate transformation in some neighborhood of (x.y) (1/2,0). (recall, that this says nothing about how far away we can go from (1/2,0) before "bad things" happen) iii) Now suppose a particle is moving vertically at (x,y)-(1/2,0) with unit speed (dxidt-0 dyldT1). Compute the time derivatives, (dt/dt, ds/dt) of (t,s) at that moment using the formulas from section 13.6 (analogous to the procedure used to derive eq. 41 42). (Answer: dtldtT-0, ds/dt 2.67) iv) You should have gotten dt/dT-0 from the previous part if you did it correctly. Relate this to the blue and red circles at the point x,y (1/2,0) Problem 9) Y u may/should use mathematica or alpha to help taking derivatives Bipolar co-ordinates are useful for describing various engineering problems (antenna/ microphone design, heat sources near a circular boundary, two parallel cylindrical thermal or electrical conductors, etc) The transformation from Bipolar co-ordinates (t,s) to Cartesian co-ordinates (x,y) is described by x-a(sinh[t])/(cosh[t]-cos[s]) y-a(sin[s)/(cosh[t-cos[s]) For purposes of this problem, set a-1 Curves of constant t and s describe "Apollonian circles" (colored red and blue) and the corresponding x.y value is the intersection of the pair of Apollonian circles for each t,s. In the figure (shamelessly stolen from wikipedia), the red circles (constant s) each pass through the two points (-1,0) and (1,0) are centered at (x.y)-(0,cot(s)) and have radius 1/sin2(s). The blue circles (constant t), are centered at (x.y)-(coth(t),0) and have radius 1/sinh(t) i) Consider the point (x.y) (1/2,0). First show the values (t,s) (1/2 Log[9], Tt) satisfy ii) Next, show that the hypotheses of the implicit function theorem are satisfied so that the co-ordinate transformation for x,y=1/20 ("Log" is natural log) we are sure to have a unique inverse function (t[(x,y)].s[(x,y)]) that satisfies the coordinate transformation in some neighborhood of (x.y) (1/2,0). (recall, that this says nothing about how far away we can go from (1/2,0) before "bad things" happen) iii) Now suppose a particle is moving vertically at (x,y)-(1/2,0) with unit speed (dxidt-0 dyldT1). Compute the time derivatives, (dt/dt, ds/dt) of (t,s) at that moment using the formulas from section 13.6 (analogous to the procedure used to derive eq. 41 42). (Answer: dtldtT-0, ds/dt 2.67) iv) You should have gotten dt/dT-0 from the previous part if you did it correctly. Relate this to the blue and red circles at the point x,y (1/2,0)