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- Let u(x,y) satisfy the following equationuxx+(x2+y2-1)uyy=0Find the region in which the equation is elliptic and the region in which the equation is hyperbolic.5.Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0.Vx,y,z=rx2+σy2+σz−2r2=c>0. a.Calculate dVdtdVdt along trajectories of the Lorenz equations (1).1. Chapter 15 Review 13: Sketch the domain D (in the xy-plane) and calculate sDf(x,y)dA.D = {0 ≤y ≤1, 0.5y2 ≤x ≤y2}, f(x,y) = ye1+x
- In Exercises 11–12, find an equation for the plane that is tangent to the given surface at the given point. 11. z = ln (x2 + y2 ), (1, 0, 0) 12. z = e^-(x2+y2), (0, 0, 1)Consider the ellipsoid V(x,y,z)=rx2+σy2+σ(z−2r)2=c>0. Determine a sufficient condition on c so that every trajectory crossing V(x, y, z) = c is directed inward.Use Green’s theorem to evaluate ∮C(ye2xy−5y)dx+ (xe2xy−2x)dy, where Cis the counterclockwise oriented boundary curve of the square with vertices at(0,0), (0,1), (1,0), and (1.1).
- E is the solid bounded by the circular paraboloid...a. Find a parametrization for the hyperboloid of one sheet x2 + y2 - z2 = 1 in terms of the angle u associated with the circle x2 + y2 = r2 and the hyperbolic parameter u associated with the hyperbolic function r2 - z2 = 1. (Hint: cosh2 u - sinh2 u = 1.) b. Generalize the result in part (a) to the hyperboloid (x2/a2 ) + (y2/b2 ) - (z2/c2 ) = 1.1. (Section 17.7) Use Stokes’ Theorem to calculate the work done by −→F (x, y, z) = ex2ˆı − 2xzˆj + xˆk in moving a particle aroundthe closed path determined by the intersection of positively oriented surface S : x + 4y + 2z = 4 and the coordinate planes.
