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Prove that an orientable compact surface SCIR3 has a differestih VEctor field without singular points if and any if s is homparnorphic to a torus Prove that an orientable compact surface SCIR3 has a differestih VEctor field without singular points if and any if s is homparnorphic to a torus
10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M 10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M. 10. Prove that a connected surface M is orientable iff there exists a nonvanishing differential 2-form on M.
5. Let ф: S1 S2 be a diffeomorphism. a. Show that S is orientable if and only if S2 is orientable (thus, orientability is preserved by diffeomorphisms). b. Let S, and S2 be orientable and oriented. Prove that the diffeomorphism ф induces an orientation in S. Use the antipodal map of the sphere (Exercise 1, Sec. 2-3) to show that this orientation may be distinct (cf. Exercise 4) from the initial one (thus, orientation itself may not be preserved by...
(8 points) Consider the graph shown below. If the graph is orientable, the orient the edges so that the resulting digraph is strongly connected. If it is not orientable, then briefly explain why. A E D F B H с
The surface factor is used to correct for difference in surface finish. Calculate the surface factor for a shaft with a ground surface finish of 45mm diameter made from a steel with an ultimate tensile strength of 1100 MPa that is subjected to a bending stress.
The acceleration of gravity at the surface of a planet is half that on the surface of Earth. If the radius of the planet is half the radius of Earth, the ratio of its mass to the mass of Earth is: 1-0.5 2-0.25 3-0.125 4- 2 Please solve it.
What is the difference between surface area, specific surface area, and specific surface area of packed bed? Explain what each represents.
Compute the surface integral over the given oriented surface: F = <0,8,x2>,hemisphere x2+y2+z2=81 , z>=0 , ,outward-pointing normal.