1.(10 pts) Prove that   ξ€Ά  ξŠξ€”ξƒ­ for any ...

  1. Home
  2. Homework Library
  3. Mathematics
  4. Differential Geometry
  5. 1.(10 pts) Prove that   ξ€Ά  ξŠξ€”ξƒ­ for any ...

QuestionQuestion

Transcribed TextTranscribed Text

1.(10 pts) Prove that   ξ€Ά  ξŠξ€”ξƒ­ for any vector field  on R 3 . 3. β‘ (7 pts) Show that d(dΞΎ)=0 for any form ΞΎ on R 3 . β‘‘(15 pts) Define #, b, and βˆ— are follows: οΌƒ: E 1( R 3 ) β†’ X- ( R 3 ) βˆ‘f idxi ↦ βˆ‘f iUi b : the inverse of # βˆ— : E 0( R 3 ) β†’E 3( R 3 ) f ↦fdx∧dy∧dz βˆ— : E 3( R 3 ) β†’E 0( R 3 ) fdx∧dy∧dz ↦f βˆ— : E 1( R 3 ) β†’E 2( R 3 ) fdx+gdy+hdz ↦fdy∧dz+gdz∧dx+hdx∧dy βˆ— : E 2( R 3 ) β†’E 1( R 3 ) ; the inverse of βˆ— : E 1 β†’ E 2 Express grad, curl, and div and the vector product Γ— in terms of d, #,βˆ—, b, and . β‘’(8 pts) Using β‘  and β‘‘, show that curl (grad f)= 0 for any differentiable function f on R 3 and div (curl V)= 0 for any vector field V on R 3 . 4.(10 pts) Prove that if  is a unit speed curve with torsion zero, then the osculating planes of  at all of its points are same. 6.(10 pts) If the spherical image of a unit speed curve   lies in a plane through the origin, prove that   is a plane curve. 7.(10pts) Prove that if ξ€…   ξ€Ά is an isometry such that   , then ξ€… is an orthogonal transformation. 8.(10 pts) If  ξƒ»βˆˆξ€“ξƒ΄   is a frame at some point of R 3 and ξ€… is an isometry of R 3 , then prove   Γ—     ×   . 9. (10 pts) Let F: R n β†’ R m be a mapping and p∈ R n . Prove that ξ€… is regular at p if and only if the Jacobian matrix of F at p has rank . 10. (10 pts) Prove that a vector field on R 3 is differentiable if and only if  ξƒͺ is differentiable for any differentiable real-valued function ξƒͺ on R 3 . 11.(10 pts) Prove that if all the osculating planes of a curve pass through a fixed point, the curve is a plane curve. 12.(10 pts) If {  } is a frame at some point of R 3 and ξ€… is an isometry, then prove ξ€…   ×     ×  13.(10 pts) Prove that if ξ€… is an isometry of R 3 , then      , where  is the orthogonal part of ξ€…. 14.(10pts) Prove that if all the principal normals of a curve pass through a fixed point, the curve is a part of a circle. 15.(10 pts) Prove that if  is a unit speed curve of constant curvature lying in a sphere, then  is a circle. 16.(10 pts) Show that every regular curve in R 3 has a unit speed reparametrization. 17.(10 pts) Let   be a unit speed curve. Show     , where ξ‚€ is the angle between the positive ξƒΌ-axis and the tangent line to the curve  measured in the counterclockwise sense. 18. (10pts) Let  be a regular curve in R 3 . Prove ξ‚―  ′ Γ— ″ ″′ ′ Γ— ″  .

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $75.00
    for this solution

    or FREE if you
    register a new account!

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

    Find A Tutor

    View available Differential Geometry 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