T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

Fachbereich Mathematik Prof. Dr. W. Trebels Dr. V. Gregoriades Dr. A. Linshaw

T E C H N I S C H E UNIVERSIT ¨ AT DARMSTADT

A

7th Tutorial Analysis II (engl.)

Summer Semester 2010

(T7.1) Consider the function f :R² →R given by

f(x, y) = x³y²+ 4xy−2x²+y³.

1. Find the equation for the tangent plane to the graph of f at the point (1,1).

2. Write down the second-order Taylor approximation T₂(x, y) to f at the point (1,1).

3. Use this polynomial to approximate the value of f(1.1,0.95).

(T7.2) Let M(n) be the set of n ×n-matrices over R, which we identify with Rⁿ

2. Consider the function

F :M(n)→ M(n), F(A) = AA^t,

where A^t is the transpose of the matrix A. Prove that F is differentiable, and compute F⁰(In), where In is the identity matrix.

Hint: Compute the directional derivativeD_BF(A) forA, B ∈ M(n), B 6= 0_n.

(T7.3) (Leibniz Formula)

Letα ∈Nⁿ0 be a multi-index α= (α₁, . . . , α_n) and U ⊂Rⁿ be an open set. Prove that

D^α(f g)(x) = X

β∈Nⁿ0, β≤α

α β

D^βf(x)D^α−βg(x) (1)

for all f, g : U → R being |α|-times partial differentiable. By β ≤ α we mean βi ≤ αi for alli∈Nⁿ0 and

α β

:= α!

β!(α−β)! :=

n

Y

i=1

α_i! β_i!(α_i−β_i)!, for all β ∈Nⁿ0, β ≤α. Hint: Use induction on the dimension n.