Calculus Answers

Prove that
Δφ(ξ) = - |ξ|^(2) (φ-hat)(ξ), ξ∈R^n,
for all φ in S, where

n
Δ = Σ (∂^2)/(∂x^2)
j=1 j

Let φ be the function on R^n defined by
φ(x) = e^(-|x|^(2)/2, x∈R
Compute (φ*φ)(x) for all x in R^n.

Let f be a nonnegative function in L^1(R^n). Prove that
|f̂(ξ)| ≤ f̂(0), ξ∈R^n

Prove that if φ and ѱ are functions in S, then so is φ*ѱ.

Prove that for all multi-indices a and all functions φ in S, x^(a)φ∈S and ∂ ^(a)φ∈S.

2. Let a and β be multi-indices such that β ≤ a. Prove that
∂^(β) x^(a) = Σ (a,β) β!x^(a - β)
β≤a

Prove that for all x in R^n and all multi-indices a,
|x^a| ≤ |x|^(|a|)

Trace r=a sin theta

Prove that for all x in R^n and all multi-indices a,
|x^a|<= |x|^(|a|)

y=x^3 - 11x^2+35x+5