School

Boston University **We aren't endorsed by this school

Course

ME 570

Subject

Mathematics

Date

Nov 13, 2023

Type

Other

Pages

4

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Question 1.2:
R1(θ)=diag(1,R(θ)):
Rotation Axis: There is no rotation around any fixed axis because the first element is 1, meaning there is no rotation around the x-
axis, y-axis, or z-axis.
Direction of Rotation: For increasing values of θ, there is only a rotation in the plane determined by the x and y axes,
counterclockwise when viewed from the positive z-axis towards the origin.
R2(θ):
Rotation Axis: This rotation is around the y-axis.
Direction of Rotation: For increasing values of θ, the rotation is counterclockwise when viewed along the positive y-axis.
R3(θ)=diag(R(θ),1):
Rotation Axis: This rotation is around the z-axis.
Direction of Rotation: For increasing values of θ, the rotation is counterclockwise when viewed along the positive z-axis.
R4(θ)=diag(−R(θ),1):
Rotation Axis: This rotation is around the z-axis.
Direction of Rotation: For increasing values of θ, the rotation is clockwise when viewed along the positive z-axis.
R5(θ)=diag(R(−θ),1):
Rotation Axis: This rotation is around the z-axis.
Direction of Rotation: For increasing values of θ, the rotation is clockwise when viewed along the positive z-axis. Note that R(−θ)
represents a counterclockwise rotation with θ in the opposite direction, so R5(θ) effectively rotates clockwise for increasing θ.
Question 3.2:

Question 3.3:
The map ϕtorus is a homeomorphism if it is bijective and continuous, and its inverse is continuous. For the given U and T,
ϕtorus is indeed bijective and continuous. However, to ensure that ϕtorus is a homeomorphism, we need to ensure that the
inverse of ϕtorus, is also continuous. The ϕtorus inverse function is continuous everywhere except when x=0 and z=0
simultaneously, where it is undefined. To handle this, we should modify the domain of U to exclude points where x=0 and
z=0.
For the torus T, at least two charts are needed in an atlas to cover it. This is because T has a non-trivial topology that cannot
be covered by a single chart. One chart covers the circle along the horizontal axis, and the other chart covers the circle along
the vertical axis. These charts are necessary to understand the torus's shape properly.
Question 3.4:

Question 3.7: