 # 1.(10 pts) Prove that      for any ...

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1.(10 pts) Prove that      for any vector field  on R 3 . 3. ①(7 pts) Show that d(dξ)=0 for any form ξ on R 3 . ②(15 pts) Define #, b, and ∗ are follows: ＃: E 1( R 3 ) → X- ( R 3 ) ∑f idxi ↦ ∑f iUi b : the inverse of # ∗ : E 0( R 3 ) →E 3( R 3 ) f ↦fdx∧dy∧dz ∗ : E 3( R 3 ) →E 0( R 3 ) fdx∧dy∧dz ↦f ∗ : E 1( R 3 ) →E 2( R 3 ) fdx+gdy+hdz ↦fdy∧dz+gdz∧dx+hdx∧dy ∗ : E 2( R 3 ) →E 1( R 3 ) ; the inverse of ∗ : E 1 → E 2 Express grad, curl, and div and the vector product × in terms of d, #,∗, b, and . ③(8 pts) Using ① and ②, 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     is an isometry such that   , then  is an orthogonal transformation. 8.(10 pts) If  ∈   is a frame at some point of R 3 and  is an isometry of R 3 , then prove   ×     ×   . 9. (10 pts) Let F: R n → R m be a mapping and p∈ R n . Prove that  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   is differentiable for any differentiable real-valued function  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  is an isometry, then prove    ×     ×  13.(10 pts) Prove that if  is an isometry of R 3 , then      , where  is the orthogonal part of . 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  is the angle between the positive -axis and the tangent line to the curve  measured in the counterclockwise sense. 18. (10pts) Let  be a regular curve in R 3 . Prove   ′ × ″ ″′ ′ × ″  .

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