MAT2007: Elementary Real Analysis II
Assignment #5
Deadline: Mar. 20
1. Suppose f (x, y) is C 1 (i.e. continuously differentiable) in a neighborhood of (0, 1) and
fy (0, 1) ̸= 0.
f (0, 1) = 0,
(i) Prove that, in a neighborhood of (0, π2 ), the equation
Z t
f x,
sin xdx = 0
0
determines a unique function t = φ(x).
(ii) Find φ′ (0).
2. Assume
x = y + f (y),
where f (0) = 0, f is C 1 and |f ′ (y)| ≤ k < 1 for −a < y < a. Show that
(i) there exists a δ > 0 such that, whenever −δ < x < δ, there is a unique C 1 function
y = y(x) satisfies the above equation and y(0) = 0.
(ii) y = y(x) is monotone for −δ < x < δ.
3. Is it possible to determine an implicit function of two variables from
xy + z ln y + exz = 1
in a neighborhood of (0, 1, 1)? Why?
4. Let P0 (x0 , y0 ) be a point in R2 with x0 > 0. Show that, in a sufficiently small neighborhood
of P0 , the system
(
u = (ex + 1) sin y
v = (ex − 1) cos y
determines inverse functions x(u, v) and y(u, v) that are continuously differentiable.
1
5. Let (x1 , . . . , xn ) and (r, θ1 , . . . , θn−1 ) be the Cartesian and spherical coordinates of Rn
respectively. That is
x1 =r cos θ1
x2 =r sin θ1 cos θ2 ,
x3 =r sin θ1 sin θ2 cos θ3 ,
······
xn−1 =r sin θ1 sin θ2 · · · sin θn−2 cos θn−1 ,
xn =r sin θ1 · · · sin θn−1 .
(i) Show that
F1 =r2 − (x21 + x22 + · · · + x2n ) = 0,
F2 =r2 sin2 θ1 − (x22 + · · · + x2n ) = 0,
······
Fn =r2 sin2 θ1 · · · sin2 θn−1 − x2n = 0.
(ii) Use the result in part (i) to calculate the Jacobian determinant
∂(x1 , . . . , xn )
.
∂(r, θ1 , . . . , θn−1 )
Hint. For the system of equations
F (x, u) = 0,
where x = (x1 , . . . , xn ), u = (u1 , . . . , un ), F = (F1 , . . . , Fn ), show that
∂(u1 , . . . , un )
∂(F1 , . . . , Fn )
n ∂(F1 , . . . , Fn )
= (−1)
.
∂(x1 , . . . , xn )
∂(x1 , . . . , xn )
∂(u1 , . . . , un )
[Why this result simplifies the calculation of the Jacobian determinant?
Note. For an upper or lower triangular matrix, its determinant equals to the product of
diagonal entries.]
6. Assume u = u(x) is determined implicitly by the system
u = f (x, y, z),
where f, g, h are C 2 functions. Find
g(x, y, z) = 0,
d2 u
du
and 2 .
dx
dx
7. Show that
x =r cos θ cos φ
y =r cos θ sin φ
z =r sin θ
are independent functions.
2
h(x, y, z) = 0,
8. Consider the linear transform from Rn → Rm ,
u = Ax,
where u = (u1 , u2 , . . . , um )t , x = (x1 , x2 , . . . , xn )t , and A = (aij ) is an m × n matrix. Prove
the following three conditions are equivalent
(i) u1 , u2 , . . . , um are dependent at a point x0 ;
(ii) u1 , u2 , . . . , um are linearly dependent;
(iii) u1 , u2 , . . . , um are dependent on Rn .
9. Suppose D is a domain in R2 , and K ⊂ D is a bounded and closed set. Assume f (x, y)
and g(x, y) are both C 1 in D, and
∂(f, g)
̸= 0
∂(x, y)
in D. Show that there are at most finitely many points in K satisfy the system
(
f (x, y) = 0,
g(x, y) = 0.
— End —
3