General geom

Question 1

Explain what is meant by saying that two bases in E3 have the same orientation

Answer 1

Two bases {e1,e2,e3} and {e1’,e2’,e3’} in V have the same orientation if the determinant of transition matrix from the first basis to the second one is positive: det T > 0

Question 2

State the Euler Theorem about rotations

Answer 2

Let P be an orthogonal operator preserving
an orientation of Euclidean space E3, i.e. operator P preserves the scalar product and orientation. Then it is a rotation operator with respect to an axis l on the angle ϕ. Every vector N directed along the axis does not change, i.e. the axis is 1-dimensional space of eigenvectors with eigenvalue 1, P(N) = N. Every vector orthogonal to axis rotates on the angle ϕ in the plane orthogonal to the axis,

Tr P = 1 + 2 cos ϕ .

The angle ϕ is defined up to a sign. Changing orientation of the Euclidean space and of the axis change sign of ϕ

Question 3

Give the definition of a differential 1-form on En

Answer 3

Differential 1-form ω on En is a function on tangent vectors of En, such that it is linear at each point:

ω(r, λv1 + µv2) = λω(r, v1) + µω(r, v2).

Question 4

Describe what is meant by a natural parameter on a curve in En

Answer 4

A natural parameter s = s(t) on the curve r = r(t) is a parameter which defines the length of the arc of the curve between initial point r(t1) and the point r(t)

Question 5

Give the definition of the curvature of a curve in En

Answer 5

The curvature of the curve in a given point is equal to
the modulus (length) of acceleration vector (normal acceleration) in natural parameterisation. Namely, let r(s) be natural parameterisation of this curve. Then curvature at every point r(s) of the curve is equal to the length of acceleration vector: 

k = |a(s)|, a(s) = d2r(s)/ds2

Question 6

Explain what is meant by saying that a differential 1-form is exact

Answer 6

1-form ω is called exact if there exists a function f such that ω = df.

Question 7

ω ? not on vector space A

Answer 7

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Question 8

ω with vector space A

Answer 8

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Question 9

length of a curve

Answer 9

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Question 10

dr(∂ϕ) and such

Answer 10

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Question 11

∫ω around C

Answer 11

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Question 12

if ω is exact then

∫ω around C ?

Answer 12

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Question 13

define curvature k(t), in both ways

Answer 13

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Question 14

angle between two vectors?

Answer 14

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Question 15

what is the natural parameter in maths

Answer 15

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Question 16

ω(v)?

Answer 16

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Question 17

shape operator?

Answer 17

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Question 18

general unit normal vector?

Answer 18

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Question 19

how to test if r is orthogonal to the surface

Answer 19

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Question 20

ω is exact if?

Answer 20

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Question 21

trig formule for sin^2x and cos^2x

Answer 21

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Question 22

linearly independent if ?

Answer 22

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Question 23

area of a parallelogram?

Answer 23

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Question 24

volume of a parallelopiped?

Answer 24

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Question 25

du ?

Answer 25

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Question 26

∂u ?

Answer 26

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Question 27

speed is increasing means?

Answer 27

(a,v) > 0

Question 28

speed is decreasing means?

Answer 28

(a,v)

Question 29

how to test if a basis is a basis?

Answer 29

if its det of transition matrix is not zero. and each vector needs to be linearly indep

Question 30

how to test if a basis is orthogonal

Answer 30

if det = +or- 1

Question 31

how does the are of a parallelogram change with choice of parametrisation

Answer 31

x by t_tau ^3

Question 32

|v x y| = ?

Answer 32

|v||y||sin(theta)|

Question 33

sin cos tan angles ?

Answer 33

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Question 34

is a curve is given under natural parametrization, whats different about the speed and v and a

Answer 34

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Question 35

x y z in polar 3d coordinates

Answer 35

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