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3247Jacobian of a Shear-Rotation Style MapFor x=u+v,\ y=u-v, compute | \partial(x,y) \partial(u,v) |.数学中等derivation未尝试面试订阅3248Polar Coordinates JacobianFor x=r\cos ,\ y=r\sin , compute | \partial(x,y) \partial(r, ) |.数学中等derivation未尝试面试订阅3249Jacobian of a Triangular MapFor x=u,\ y=u+2v, compute | \partial(x,y) \partial(u,v) |.数学中等derivation未尝试面试订阅3250Jacobian of a Simple Nonlinear MapFor x=u 2-v,\ y=u+v 2, compute \partial(x,y) \partial(u,v) symbolically.数学中等derivation未尝试面试订阅3261Area of an Ellipse from the Unit DiskThe map x=2u,\ y=3v sends the unit disk u 2+v 2\le 1 to an ellipse in the (x,y)-plane. What is the area of that ellipse?数学中等derivation未尝试面试订阅3262Area of a Parallelogram from a Unit SquareThe map x=u+v,\ y=u-v sends the unit square 0\le u,v\le 1 to a parallelogram. What is its area?数学中等derivation未尝试面试订阅3263Integral After a Simple ScalingUse the change of variables x=2u,\ y=3v to compute \iint R 1\,dx\,dy, where R is the image of the rectangle 0\le u\le 1,\ 0\le v\le 2.数学中等derivation未尝试面试订阅3264Area Scaling Under a Triangular MapThe map x=u,\ y=u+2v sends the unit square to a parallelogram. Compute the image area.数学中等derivation未尝试面试订阅3265Area of a Polar WedgeUse the polar Jacobian to compute the area of the region 0\le r\le 2,\ 0\le \le /3.数学中等derivation未尝试面试订阅3291Exponential Decay After Two Time UnitsSolve x'(t)=-3x(t) with x(0)=12, and evaluate x(2).数学简单derivation未尝试面试订阅3293Cooling Model Over Half a UnitSolve x'(t)=-2x(t) with x(0)=10, and evaluate x( 1 2 ).数学简单derivation未尝试面试订阅3296Linear ODE with Constant ForcingSolve x'(t)+2x(t)=6 with x(0)=1, and simplify the solution.数学中等derivation未尝试面试订阅3306Simple Harmonic Oscillator SolutionSolve y''+4y=0 with y(0)=1 and y'(0)=0.数学中等derivation未尝试面试订阅3316Decay of the First Sine ModeFor the heat equation u t=1u xx on 0<x< with zero boundary conditions, the initial condition is u(x,0)=\sin(1x). What is u(x,t)?数学中等derivation未尝试面试订阅3321Two-Mode Initial ConditionFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)+2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅3323Three-Mode Heat ProfileFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=2\sin(1x)+1\sin(2x)+1\sin(3x), write u(x,t).数学中等derivation未尝试面试订阅3325Alternating Signs in Initial HeatFor u t=1u xx on 0<x< with zero boundary conditions and initial condition u(x,0)=1\sin(1x)-2\sin(2x), write u(x,t).数学中等derivation未尝试面试订阅4916Infer Normal Parameters From VaR And ES 1A desk assumes losses are normal with mean mu and standard deviation sigma. At alpha=0.95 it uses VaR = mu + z*sigma and ES = mu + k*sigma with z=1.645 and k=2.063. If the reported VaR is 7.58 and ES is 9.252, what are mu and sigma?数理金融中等数值题未尝试面试订阅4917Infer Normal Parameters From VaR And ES 2A normal-loss desk uses alpha=0.99 with VaR = mu + z*sigma and ES = mu + k*sigma, where z=2.326 and k=2.665. If VaR is 7.478 and ES is 8.495, what are mu and sigma?数理金融中等数值题未尝试面试订阅4918Infer Normal Parameters From VaR And ES 3A risk report uses alpha=0.975 with z=1.96 and k=2.338 in the formulas VaR = mu + z*sigma and ES = mu + k*sigma. If VaR is 11.8 and ES is 13.69, what are mu and sigma?数理金融中等数值题未尝试面试订阅