(2) Let f (x, y

1. hw 3 Deadline:10/07/5:00pm

(1) Bartle’s book exercise: do Page 356: 39C.

(2) Let f (x, y) = x³− 3xy² for (x, y) ∈ R². Let p = (1, 1) and u = (cosπ 3, sinπ

3) and v = (cosπ

6, sinπ

6). Evaluate (a) Duf (p), Dvf (p)

(b) D²_uuf (p), D_uv² f (p), D_vu² f (p) and D_vv² f (p).

(3) Find all the first and second partial derivatives of f (x, y) = x^y for (x, y) ∈ D = {(x, y) : x >

0, x 6= 1}.

(4) Find the directional derivatives of each of the following functions as indicated:

(a) xyz − xy − yz − zx + x + y + z at (2, 2, 1) in the direction of (2, 2, 0).

(b) xz²+ y²+ z³ at (1, 0, −1) in the direction of (2, 1, 0).

(5) Find all partial derivatives through the third order of the following functions f (x, y) = cosh(xy), (x, y) ∈ R².

(6) Let D₁, · · · , D_m be subsets of Rⁿ and f_i : D_i → R be functions for 1 ≤ i ≤ m. Let D = Tm

i=1D_i. Suppose that int(D) is nonempty. We define a vector valued function f : D → R^m by

f (p) = (f₁(p), f₂(p), · · · , f_m(p)), p ∈ D.

If u is a unit vector in Rⁿ, we define the directional derivative of f at p along u to be Duf (p) = lim

t→0

1

t(f (p + tu) − f (p))

when the limit exists. In calculus, we learn the following Theorem:

Theorem 1.1. Let F : (a − d, a + d) → R^m be a vector valued function with F = (F1, · · · , Fm) for Fi : (a − d, a + d) → R. Then lim_t→aF(t) exists if and only if limt→aFi(t) exist for 1 ≤ i ≤ m. In this case

t→alimF(t) =

t→alimF₁(t), · · · , lim

t→aF_m(t) . (a) Show that int(D) =Tm

i=1int(D_i).

(b) By (a), we know that p is an interior point of D if and only if p is an interior point of Di for all 1 ≤ i ≤ m. Let u be a unit vector in Rⁿ. Using Theorem 1.1 to show that Duf (p) exists if and only if Dufi(p) exists for 1 ≤ i ≤ m and that

D_uf (p) = (D_uf₁(p), · · · , D_uf_m(p)).

(We have done this proof in class. We ask you to do this again.)

(c) Let f1(x, y) = axⁿ+ by^m with D1 = R² and f2(x, y) = 2xe^y2+ 3y with D2 = R² and f3(x, y) = 2^x_y + 3_x^y with D3 = {(x, y) : xy 6= 0}. Here a, b, n, m are real numbers. Let D = D₁∩ D2∩ D3 and f : D → R³ be the function f = (f₁, f₂, f₃), i.e.

f (x, y) =



axⁿ+ by^m, 2xe^y2+ 3y, 2x y + 3y

x

 . Compute fx, fy when they exist.

(7) (Laplace Equation) Let U = Rⁿ\ {0} for n ≥ 3. Define a function f : U → R by f (x₁, · · · , x_n) = 1

(x²₁+ · · · + x²_n)^(n−2)/2. (a) Prove that U is an open subset of Rⁿ.

(b) Prove that fx_i exists on U for 1 ≤ i ≤ n. Also find fx_i. (c) Prove that fx_ix_j exists on U for 1 ≤ i, j ≤ n. Also find fx_ix_j. (d) Prove that fx₁x₁+ · · · + fx_nx_n= 0 holds on U.

1

2

(8) (Heat Equation) Let U = {(x, y) ∈ R²: y > 0}. Define a function f : D → R by f (x, y) = 1

√ye^{−(x−a)24y} , (x, y) ∈ D.

By hw 1, U is an open set.

(a) Find D ⊂ U so that f_x(x, y), f_y(x, y), f_xx(x, y) exist for all (x, y) ∈ D. Is it true that D = U ? Is D open? Find int(D) if D is not open.

(b) Show that the equation f_y = f_xxholds on int(D).

If you want to practice more exercises on calculating the partial derivatives, you can check Courant and John volume 2-1:

(1) Page 30-31: Exercise 1.4 a: 1,2, 3, 4, 5, 6, 7, 8.

(2) Page 39: Exercise 1.4 d: 1,2,3.