For the following exercises, find the area of the surface obtained by rotating the given curve about the x -axis. 124. Find the surface area generated by revolving x = t 2 , y = 2 t 2 , 0 ≤ t ≤ 1 about the y -axis.
For the following exercises, find the area of the surface obtained by rotating the given curve about the x -axis. 124. Find the surface area generated by revolving x = t 2 , y = 2 t 2 , 0 ≤ t ≤ 1 about the y -axis.
This is advanced mathematics question that need detailed solutions
Question:
Let F be a field. Prove that F contains a unique smallest subfield, called the prime subfield, which is
isomorphic to either Q or Zp for some prime p.
Instructions:
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Begin by identifying the identity element 1 € F.
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Use the closure under addition and inverses to build a subring.
•
•
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Show that either the map ZF or Q →F is an embedding.
Prove minimality and uniqueness.
Discuss the characteristic of a field and link it to the structure of the prime subfield.
Topic: Group Theory | Abstract Algebra
Question:
Let G be a finite group of order 45. Prove that G has a normal subgroup of order 5 or order 9, and describe
the number of Sylow subgroups for each.
Instructions:
•
Use Sylow's Theorems (existence, conjugacy, and counting).
•
List divisors of 45 and compute possibilities for n for p = 3 and p = 5.
Show that if n = 1, the subgroup is normal.
Conclude about group structure using your analysis.
University Calculus: Early Transcendentals (4th Edition)
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY