Exam questions Flashcards
Explain what is meant by saying that two bases in E3 have the same orientation
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
State the Euler Theorem about rotations
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
Give the definition of a differential 1-form on En
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).
Describe what is meant by a natural parameter on a curve in En
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)
Give the definition of the curvature of a curve in En
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
Explain what is meant by saying that a differential 1-form is exact
1-form ω is called exact if there exists a function f such that ω = df.
Orthogonal operator preserves….?
scalar product
Rotation operator preserves….?
orientation. detP=1
Operator is orthogonal if…
A^TA=1
Same orientation if…
detT>0
Opposite orientation if….
detT
An ordered triple is a basis if…
vectors are linearly independent
or
if non degenerate i.e detT =/= 0
Orthonormal basis means
scalar product holds
