QuestionQuestion

1. Let D ⊆ D ⊆ Rⁿ denote rectangles. Let f: D --> R. Assume F(x) ≥ 0, x ∈ D.
i) Assume ∫D₁ F does not exist. Show ∫D F does not exist.
ii) Assume ∫D F exists. Show ∫D₁ exists.

2. Let D = [0, 1] x [0, 1] ⊂ R² and assume f: D --> R is continuous. Define S ⊆ R³ by {(x,y,z) z = F(x,y), (x,y) ∈ D}. Show S has content 0.

3. Consider the transformation from R² --> R³ given by u = x² + y², v = xy. By the inversion theorem there exists an inverse transformation x = ψ(u,v) defined by F(u,v) in a neighborhood of (5,2) which maps (u,v) = (5,2) into (x,y) = (1,2).
i) Calculate ∂ψ/∂u(5,2).
ii) Find explicit functions x = ϕ(u,v), y = ψ(u,v)

4. Let D ⊆ Rⁿ be a rectangle. Let P, Q denote partitions of D. Let f: D --> R. Show L(P, F) ≤ u(Q, f).

5. Let Aₙₓₙ denote an invertible matrix with real entries. Define F: Rⁿ --> Rⁿ by F(x) = AX. Assume, a,b ∈ Rⁿ and F(a) = b, then the inversion theorem applies. Calculate the first two terms in the sequence, q1(x), q2(y) and show clearly q2(y) = q(y) = A⁻¹ y.

6. Let Aₙₓₙ denote a matrix with real entries. Assume A is positive definite and let λ be an eigenvalue of A. Prove that λ > 0.

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Functions and Transformations
    $33.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematics - Other Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats