1. Quiz 10 Consider the system of equations
f1(x1, · · · , xn, u1, · · · , um) = c1
... fm(x1, · · · , xn, u1, · · · , um) = cm
where f1, · · · , fm are real valued C1-functions on an open set U ⊂ Rm+n. Suppose that the above system has a solution at p. We assume further that the determinant
(1.1)
∂f1
∂u1 · · · ∂f1
∂um ... . .. ...
∂fm
∂u1 · · · ∂fm
∂um
is nonzero at p. In class, we have mentioned that in a neighborhood of p, u1, · · · , um can be solved uniquely in terms of x1, · · · , xn. Use this fact to study the following problems.
(1) Can the equation (x2+ y2+ 2z2)1/2= cos z be solved uniquely for y in terms of x, z near (0, 1, 0)? For z in terms of x and y?
(2) Can the system of equations
xy2+ xzu + yv2= 3 u3yz + 2xv − u2v2= 2
be solved for u, v as functions of x, y, z near (1, 1, 1, 1, 1)?
(3) Can the system of equations
xu + yvu2 = 2 xu3+ y2v2 = 2
be solved uniquely for u, v as functions of x and y near (1, 1, 1, 1)? Compute ux(1, 1).
(4) Let us try to prove the case when m = 2 and n = 3, i.e. study the solvability of the following system of equations
f1(x1, x2, x3, u1, u2) = c1
f2(x1, x2, x3, u1, u2) = c2
Here f1, f2 : E ⊂ R5 → R are C1 functions defined on an open subset E of R2. Suppose p = (x0, y0, z0, u0, v0) is a solution to the above equation. Suppose that
∆ =
∂f1
∂u1(p) ∂f1
∂u2(p)
∂f2
∂u1
(p) ∂f2
∂u2
(p)
6= 0.
(a) Define a function f : E ⊂ R5 → R5 by
f (x1, x2, x3, u1, u2) = (x1, x2, x3, f1(x1, x2, x3, u1, u2), f2(x1, x2, x3, u1, u2)).
Show that Jf(p) = ∆.
1
2
(b) By inverse function theorem f is invertible in a neighborhood of p. We choose U = (x0− δ, x0+ δ, y0− δ, y0+ δ, z0− δ, z0+ δ, u0− δ, u0+ δ, v0− δ, v0+ δ)
(think that why we can choose such U ) and V = f (U ) such that f : U → V is a bijection so that g = f−1 : V → U is also C1. Write g = (g1, g2, g3, g4, g5). Show that
g1(v1, v2, v3, v4, v5) = v1 g2(v1, v2, v3, v4, v5) = v2
g3(v1, v2, v3, v4, v5) = v3
f1(v1, v2, v3, g4(v1, v2, v3, v4, v5), g5(v1, v2, v3, v4, v5)) = v4 f2(v1, v2, v3, g4(v1, v2, v3, v4, v5), g5(v1, v2, v3, v4, v5)) = v5 for any (v1, v2, v3, v4, v5) ∈ V.
(c) Define two functions ψ, φ : (x0− δ, x0+ δ, y0− δ, y0+ δ, z0− δ, z0+ δ) → R by ψ(x1, x2, x3) = g4(x1, x2, x3, c1, c2)
φ(x1, x2, x3) = g5(x1, x2, x3, c1, c2).
Prove that ψ, φ are both C1-functions such that
f1(x1, x2, x3, ψ(x1, x2, x3), φ(x1, x2, x3)) = c1
f2(x1, x2, x3, ψ(x1, x2, x3), φ(x1, x2, x3)) = c2.
In other words, u1 = ψ(x1, x2, x3) and u2 = φ(x1, x2, x3) are solutions to the system of equations for any (x1, x2, x3) ∈ (x0−δ, x0+δ, y0−δ, y0+δ, z0−δ, z0+δ).
Furthermore, show that
ψ(x0, y0, z0) = u0 φ(x0, y0, z0) = v0. (d) Show that
∂f1
∂x1
+ ∂f1
∂u1
∂u1
∂x1
+ ∂f1
∂u2
∂u2
∂x1
= 0
∂f2
∂x1 + ∂f2
∂u1
∂u1
∂x1 + ∂f2
∂u2
∂u2
∂x1 = 0.
Find ∂u1
∂x1
(q) and ∂u2
∂x1
(q) where q = (x0, y0, z0).
(e) Compute ∂u1
∂xi(q) and ∂u2
∂xi(q) for all 1 ≤ i ≤ 3.
