Let F : V → R be the function F (u

1. Homework 2

(1) Let V = C([0, 1]) be the space of real valued continuous functions on [0, 1] equipped with the sup-norm

kuk_∞= sup

x∈[0,1]

|u(x)|, u ∈ V.

Let F : V → R be the function F (u) =

Z ₁

0

u(x)³dx, u ∈ V.

Let u₀∈ V be a nonzero element in V.

(a) Let T : V → R be the map T (h) =

Z 1 0

3u₀(x)²h(x)dx.

Prove that T is a bounded linear map.

(b) Assume that khk∞< ku₀k_∞. Prove that

|F (u₀+ h) − F (u0) − T (h)| ≤ 4ku0k_∞khk²_∞.

(c) Prove that f is differentiable at u0 and find DF (u0). (In fact, we do not need to assume that u₀ = 0.)

(d) Let f : R → R be any smooth function and F : V → R, F (u) =

Z 1 0

f (u(x))dx.

Is F differentiable at u ∈ V ? If it is differentiable at u, prove or disprove that DF (u)(h) =

Z 1 0

f⁰(u(x))h(x)dx.

(Hint: mean value theorem and the continuity of f⁰(t),). Do the case when f : R → R is a polynomial (function).

(2) Let U be an open subset of Rⁿ and f : U → R^m be a function. Let u be a nonzero vector in Rⁿ. Recall that the directional derivative¹of f at p ∈ U along u is defined to be

Duf (p) = lim

t→0

1

t(f (p + tu) − f (p)).

(a) Let u be any vector in Rⁿ. Show that if f is differentiable at p, then the direc- tional derivative of f at p along u exists and equals Df (p)(u), i.e.

(1.1) D_uf (p) = Df (p)(u).

This implies that if f is differentiable at p, then D_uf (p) is a linear map from Rⁿ to R^m in u.

(b) Verify that (1.1) holds in the case when f : R² → R², f (x, y) = (x² − y², 2xy) and p = (2, 1) and u = (1, 1).

(3) Prove that the function

f : R² → R, f (x, y) =p|xy|

is not differentiable at (0, 0).

1In Advanced Calculus I, we require u to be a unit vector. In fact, we can drop this assumption.

1

2

(4) Prove that if α > 1/2 the function f (x, y) =

(|xy|^αlog(x²+ y²) if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0)

is differentiable on R². (5) Prove that the function

f (x, y) =







x³− xy²

x²+ y² if (x, y) 6= (0, 0) 0 if (x, y) = (0, 0)

is continuous on R², has first order partial derivatives everywhere on R² but is not differentiable at (0, 0).

(6) (Product Rule) Let U be an open subset of Rⁿ and f, g : U → R be functions.

Suppose that f and g are differentiable at p. Show that (1.2) D(f g)(p) = g(p)Df (p) + f (p)D(g)(p).

Let f (x, y) = x − y, g(x, y) = x²− y², p = (1, 1). Verify (1.2) holds for the given f, g and p.

(7) (Quotient Rule) Let U be an open subset of Rⁿ and f : U → R be a function.

Suppose that f is differentiable at p with f (p) 6= 0.

(a) Show that there exists δ > 0 such that f (p + h) 6= 0 for any khk < δ.

(b) Prove that Df (p)(h)/khk is bounded for all h 6= 0.

(c) Let T = −(Df )(p)/f²(p). Show that if khk < δ, 1

f (p + h) − 1

f (p)− T (h) = f (p) − f (p + h) + Df (p)(h) f (p)f (p + h)

+(f (p + h) − f (p))Df (p)(h) f²(p)f (p + h) . (d) Prove that 1/f (x) is differentiable at x = p and

D 1 f



(p) = −Df (p) f²(p).

(e) Suppose g : U → R is a function differentiable at a. Prove that D g

f



(p) = f (p)Dg(p) − g(p)Df (p)

f²(p) .

Computational Problems

(8) Let f (x, y, z) = (x − z, x + z), g(x, y, z) = (xyz, x²− y²), for (x, y, z) ∈ R³ and p = (1, 1, 1). Find D(f + g)(p) and D(3f − 2g)(p).

Let V and W be n and m dimensional real vector spaces. Suppose that β = {v₁, · · · , v_n} and γ = {w₁, · · · , w_m} are basis for V and for W respectively. For each v ∈ V, we write

x = x1v1+ x2v2+ · · · + xnvn

3

for x₁, · · · , x_n∈ R. The n × 1 matrix [v]β =









 x₁ x2

... xn











is called the coordinate (matrix)

representation of v with respect to β. If T : V → W is a linear map, we denote [T ]^γ_β =[T (v₁)]γ [T (v2)]γ · · · [T (vn)]γ .

The m × n real matrix [T ]^γ_β is called the matrix representation of T with respect to β and γ. One can easily verify that

[T (v)]_γ = [T ]^γ_β[v]_β.

Let f : U → R^m be a function defined on an open subset U of Rⁿ and p ∈ U be a point. We write f = f (x1, · · · , xn). If f is differentiable at p, then the matrix representation of Df (p) with respect to the standard basis for Rⁿ and R^m is given by

[Df (p)]γ=

 ∂f

∂x1

(p)



γ

 ∂f

∂x2

(p)



γ

· · ·  ∂f

∂xn

(p)



γ

 . For example, if f (x, y, z) = (f (x, y, z), g(x, y, z)), then

∂f

∂x = (fx, gx), ∂f

∂y = (fy, gy), ∂f

∂z = (fz, gz) and hence

 ∂f

∂x



γ

=f_x g_x



,  ∂f

∂y



γ

=f_y g_y



,  ∂f

∂z



γ

=f_z g_z



which gives

[Df (x, y, z)] =f_x(x, y, z) f_y(x, y, z) f_z(x, y, z) gx(x, y, z) gy(x, y, z) gz(x, y, z)

 .

(9) Let f : R³ → R², f (x, y, z) = (x⁴y, xe^z). Compute Df (x, y, z). Find the matrix representation of Df with respect to the standard basis for R³ and R², i.e. find [Df (x, y, z)]^γ_β where β and γ are the standard basis for R³ and R² respectively.

Check the equation

[Df (x, y, z)(h)]_γ = [Df (x, y, z)]^γ_β[h]_β to see if your calculation is correct.

(10) Let f : R³ → R, f(x, y, z) = e^x2+y2+z2. Compute Df (x, y, z). Find the matrix representation of Df (x, y, z) with respect to the standard basis for R³ and R, i.e.

find [Df (x, y)]^γ_β where β and γ are the standard basis for R³ and R respectively.

Also Check

[Df (x, y, z)(h)]γ = [Df (x, y, z)]^γ_β[h]_β.

(11) Let U = R²\ {(x, y) ∈ R² : x = 0} and f : U → R², f (x, y) = (xy, y/x).

(a) Compute Df (x, y).

(b) Compute the matrix representation of Df (x, y) with respect to the standard basis for R², i.e. find [Df (x, y)]β where β is the standard basis for R².

(c) Compute the matrix representation of Df (x, y) with respect to the basis γ = {(1, 0), (1, 1)} i.e. find [Df (x, y)]_γ.

4

(d) Check

[Df (x, y)(h)]_γ = [Df (x, y)(h)]_γ[h]_γ.