## Transcribed Text

1.(10 pts) Prove that ξ ξ΄
ξΆ
ξξξΌξ ξξξ for any vector field ξ 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 β are follows:
οΌ: E
1( R
3 ) β X- ( R
3 )
βf idxi β¦ βf iUi
b : the inverse of #
β : E
0( R
3 ) βE
3( R
3 )
f β¦fdxβ§dyβ§dz
β : E
3( R
3 ) βE
0( R
3 )
fdxβ§dyβ§dz β¦f
β : E
1( R
3 ) βE
2( R
3 )
fdx+gdy+hdz β¦fdyβ§dz+gdzβ§dx+hdxβ§dy
β : E
2( R
3 ) βE
1( R
3 ) ; the inverse of β : E
1 β E
2
Express grad, curl, and div and the vector product Γ in terms of d, #,β, 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 ξ is a unit speed curve with torsion zero, then the osculating
planes of ξ at all of its points are same.
6.(10 pts) If the spherical image of a unit speed curve ξξξ· ξ
lies in a plane through
the origin, prove that ξξξ· ξ
is a plane curve.
7.(10pts) Prove that if ξ
ξ ξ
ξΆ
is an isometry such that ξ
ξ ξ
ξξ½, then ξ
is an
orthogonal transformation.
8.(10 pts) If ξΊξ ξ»βξξ΄ ξξ
ξΆξ
is a frame at some point of R
3 and ξ
is an isometry of
R
3
, then prove ξ
ξ ξξΊ Γ ξ» ξ
ξ ξξ·ξ«ξ²ξ
ξ
ξ
ξ ξξΊξ
Γ ξ
ξ ξξ»ξ
.
9. (10 pts) Let F: R
n β R
m
be a mapping and pβ R
n
. Prove that ξ
is regular at p
if and only if the Jacobian matrix of F at p has rank ξ².
10. (10 pts) Prove that a vector field on R
3
is differentiable if and only if ξ ξͺξ is
differentiable for any differentiable real-valued function ξͺ 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 { ξξ©ξ΅ξξ©ξΆ } is a frame at some point of R
3 and ξ
is an isometry, then
prove ξ
ξξ©ξ΄ξ
ξ
ξ ξξ©ξ΅ξ
Γ ξ
ξ ξξ©ξΆξ
ξ ξξ·ξ«ξ²ξ
ξ
ξ©ξ΄ ξ©ξ΅Γ ξ©ξΆ
13.(10 pts) Prove that if ξ
is an isometry of R
3
, then ξ
ξ ξξΊξ΄ξ
ξ ξξξΊξ
ξξ΄ξ
, where ξ 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 ξ is a unit speed curve of constant curvature lying in a
sphere, then ξ is a circle.
16.(10 pts) Show that every regular curve in R
3 has a unit speed reparametrization.
17.(10 pts) Let ξξξ· ξ
be a unit speed curve. Show ξ¦ ξ
ξ¨ξ·
ξ¨ξ€
, where ξ€ is the angle
between the positive ξΌ-axis and the tangent line to the curve ξ measured in
the counterclockwise sense.
18. (10pts) Let ξ be a regular curve in R
3
. Prove ξ― ξ ξξβ² Γ ξβ³ξ
ξβ³β²ξ ξβ² Γ ξβ³
ξ΅
.

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.