Multivariable Cal(4Questions)

znfgreLv
timer Asked: Oct 30th, 2015

Question Description

do Question#3,4,5 in hw.pdf and the question in the picture.

It should be easy

hw.pdf 


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Analysis and Optimization. HW set 6. Explain and justify wherever possible. 1. (not for credit). 2.7.1,2.7.2,2.7.4,2.7.6 2. 2.7.9 3. Consider the subset S of R3 defined by: x2 + 2y 2 + z 4 − xy − 2yz = 1 Show that there is a C 1 function g : R2 → R defined near (1, 1) and C 1 such that for (x, y, z) close to (1, 1, 1), (x, y, z) ∈ S if and only if z = g(x, y). Compute ∇g at (1, 1). 4. Consider the nonlinear system of equations:  x + y + txy + t2 x4 =2 2 2 x + 2y + sin(t)(x − 2y ) = 3 where x, y are unknowns and t is a parameter. (a) Find the solution (x0 , y0 ) when t = 0. (b) Show that there are C 1 functions g1 , g2 defined near 0 s.t. (x, y) = (g1 (t), g2 (t)) is a solution of this system. Compute g10 (0), g20 (0). 5. (Spherical coordinates). Consider f : R3 → R3 given by: f (r, θ, φ) = (r cos θ sin φ, r sin θ sin φ, r cos φ) Evaluate Jac(f ). Show that there is a ball U around (2, π3 , π2 ) such that f is a bijection √ between U and f (U ). Evaluate Jac((f|U )−1 ) at (1, 3, 0). 1 4. Give sufficient conditions on f and g to ensure that the equations u = f(x, y), v = g(x, y) can be solved for x and y locally. Show that if the solutions are x = F(u, v), y = G(u, v), and J denotes the Jacobian determinant of f and g w.r.t. u and v, then ar 1 ag ag 1 ag au лау Jax au
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