What is Example: Definition and 877 Discussions

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  1. Geofleur

    A Pathological example of a Principle Fiber Bundle?

    In Hassani's Mathematical Physics, a principal fiber bundle is defined as shown below. I wanted to see if there is a way to view a tangent bundle as a PFB, even if the resulting structure would have to be globally trivial, so I came up with this idea: Let ##P = {\rm I\!R} \times {\rm l\!R}##...
  2. Math Amateur

    MHB Matrices of Linear Transformations .... Poole, Example 6.76 ....

    I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ... I need some help in order to fully understand Example 6.76 ... ... Example 6.76 reads as follows: My question or...
  3. R

    Probability Example: A vs B | Competing Software Companies

    Homework Statement I was reading this example in a book... " Two competing software companies are after an important contract. Company A is twice as likely to win this competition as company B. Hence, the probability to win the contract equals 2/3 for A and 1/3 for B." Homework EquationsThe...
  4. Lolaamaigatti04

    What would be a real-life example of the ideal gas law?

    Homework Statement What is a real-life example of the ideal gas law? Homework Equations PV = nRT (Pressure x volume = number of moles x the gas constant x temperature in Kelvin) The Attempt at a Solution https://www.reference.com/science/ideal-gas-law-used-everyday-life-3dacbd6ebd3b5949...
  5. V

    Proofs Homework: Explain Why m+n≠10 Example Not Sufficient

    Homework Statement Suppose you are asked to prove that for all integers m and n, m+n≠10. You give the example m=1 and n=2. Why is this not sufficient? Homework EquationsThe Attempt at a Solution I can't quite understand why it is not sufficient? Could someone please explain to me why is it...
  6. Marco Masi

    Example of a technology that was long underestimated?

    I'm wondering how and in what time frame new technologies emerged and found application in the history of science. I could make several examples of those which suddenly and unexpectedly changed the world, but not the vice-versa. In particular, I'm wondering which didn't find an application or...
  7. Math Amateur

    MHB Dual Spaces .... Friedberg et al, Example 4, Section 2.6

    I am reading the book: "Linear Algebra" by Stephen Friedberg, Arnold Insel, and Lawrence Spence ... and am currently focused on Section 2.6: Dual Spaces ... ... I need help with an aspect of Example 4, Section 2.6 ... Example 4, Section 2.6 reads as follows: (see below for details of Section...
  8. nomadreid

    I A question about a popular example of expansion

    From the outset let me assert that I am not putting into question the expansion of space. Redshifts and all that. But there is an example which one often reads in popular astronomy articles which appears at first glance to be faulty, although I suspect it is my reasoning that is faulty, and I...
  9. Math Amateur

    MHB Smooth Structures .... Dundas, Example 2.2.6 .... ....

    I am reading Bjorn Ian Dundas' book: "A Short Course in Differential Topology" ... I am focused on Chapter 2: Smooth Manifolds ... ... I need help in order to fully understand Example 2.2.6 ... ... Example 2.2.6 reads as follows: My questions are as follows: Question 1 In the above text...
  10. Math Amateur

    MHB Graphs of Functions Covering Unit Circle .... Hubbard & Hubbard, Example 3.1.5 .... ....

    I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 3.1: Manifolds ... I need some help in order to understand Example 3.1.3 ... ... Example 3.1.3 reads as follows:In...
  11. jim mcnamara

    Example vue.js application - nice periodic chart

    Chart: https://periodicity.io/ Kadin Zhang's code on github: https://periodicity.io/ edit: https://github.com/kadinzhang/Periodicity The app is very useful for beginning chemistry students, IMO, but is even better for learning the vue.js toolset for web development. Code example itself is...
  12. Math Amateur

    MHB Directional Derivative Example .... SHifrin, Ch.3, Section 1, Example 3 ....

    I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Section 3.1 Partial Derivatives and Directional Derivatives ... I need some help with Example 3 in Chapter 3, Section 1 ... Example 3 in Chapter 3, Section 1 reads as follows:In the above text we read...
  13. Math Amateur

    MHB Pointwise Convergence .... Abbott, Example 6.2.2 (iii) .... ....

    I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ... I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on pointwise convergence... I need some help to understand the 'mechanics' of Example 6.2.2 (iii) ... Example 6.2.2 reads as...
  14. Math Amateur

    MHB Directional Derivatives .... Apostol, Section 12.2, Example 4 ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.2: The Directional Derivative ... ...I need help with part of Example 4, Section 12.2 ...Section 12.2, including the...
  15. Math Amateur

    MHB Monoids .... Monos and Injective Maps .... Awodey, Example 2.3, Page 30 ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 2: Abstract Structures ... ... I need some help in order to fully understand Awodey Example 2.3, Chapter 2 ... ... Awodey Example 2.3, Chapter 2 reads as follows: In the Example above Awodey writes the...
  16. Math Amateur

    MHB Rel as a Category .... Awodey Section 1.4, Example 4, Pae 7 ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ... I need some further help in order to fully understand some aspects of the definition of the category Rel ... ... The definition of the category Rel ... reads as...
  17. Math Amateur

    MHB Diagram Chasing .... Simmons Example 2.1.2 .... .... very basic question ....

    I am reading the book: "An Introduction to Category Theory" by Harold Simmonds and am currently focused on Section 2.1: Diagram Chasing ... I need some help in order to fully understand Example 2.2.1 (b) on page 36 ... ... Example 2.1.2 reads as follows...
  18. Z

    Continuous random Var range and example

    Homework Statement Why we use range with continuous random and why is time continuous var and why we associate a range with it? Homework Equations Theoreticl topic The Attempt at a Solution Hi, I can't understand about the continuous random var and its range. It says that measurable values are...
  19. E

    B Calculate Proper Time for Two Events in Special Relativity

    I am refreshing myself on a bit of special relativity and I remember kind of taking the invariance of the spacetime interval for granted the first time. When thinking about it and running some examples I started to get confused so I decided to come here. All of the books I've read always have...
  20. Math Amateur

    I Product of Categories .... Product of Groups as an example ...

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition of...
  21. Math Amateur

    MHB How can the product of two categories help us understand groups?

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition...
  22. Math Amateur

    MHB The Category of Pointed Sets .... Awodey Example 1.8, Page 17 ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.6 Constructions on Categories ... I need some further help in order to fully understand some further aspects of Example 1.8, Page 17 ... ... Example 1.8, Page 17 ... reads as follows:I find the...
  23. Math Amateur

    MHB Understanding Monoids as Categories: Awodey Section 1.4 Example 13

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ... I need some further help in order to fully understand some further aspects of Section 1.4 Example 13 ... Section 1.4 Example 13 reads as follows: In the above text by...
  24. Math Amateur

    MHB Monoids as Categories .... Awodey Section 1.4, Example 13 ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.4 Examples of Categories ... I need some help in order to fully understand some aspects of Section 1.4 Example 13 ... Section 1.4 Example 13 reads as...
  25. BeyondBelief96

    David Morin's Rope around a Pole example

    1. A rope wraps an angle θ around a pole. You grab one end and pull with a tension T0. The other end is attached to a large object, say, a boat. If the coefficient of static friction between the rope and the pole is µ, what is the largest force the rope can exert on the boat, if the rope is to...
  26. A

    Timoshenko, Strength of materials example solution

    Hi everyone, I am new here, so please be kind if I make mistakes on this forum, I will try to learn fast. I need help in explaining the solution of one of the examples from Timoshenko's Strength of Materials book. The solution is provided in the book, but I did not get it, no matter at which...
  27. Math Amateur

    MHB Maximal Ideal .... Bland - AA - Example 2, Section 3.2.12 .... ....

    I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with the proof of Example 2, Section 3.2.12, pages 147 to 148 ... ... Example 2, Section 3.2.12 reads as follows...
  28. F

    I Real life example of the energy contained at E=γmc^2

    We've just begun studying about relativity, and I find it amazing that bodies have the energy of E=γmc^2. Even at rest they have E=mc^2. But where exactly is this energy present in real life? For example the keyboard I am currently typing this post with has a huge amount of energy, according to...
  29. Math Amateur

    MHB Endomorphism Rings .... Bland, Example 7, Section 1.1 .... ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 1.1 Rings and need some help to fully understand the proof of part of Example 7 on page 10 ... ... Example 7 on page 10 reads as follows:In the above example from Bland we read the following: " ... ...
  30. T

    Constructing a Non-Subspace in R^2 with Closed Addition and Additive Inverses

    Hi, I hope you guys help me with this exercise in the book "Linear Algebra Done Right" Homework Statement " Give an example of a nonempty subset ##U## of ##R^2## such that ##U## is closed under addition and under taking additive inverses (meaning ##−u## ##∈## ##U## whenever ##u## ##∈## ##U##)...
  31. G

    I Need a practical example of E=mc^2 with real numbers

    Need a practical example of E=MC2 with real numbers Ok, so I understand that Energy = Mass of an Object * Speed of light square, if we must convert this to numbers, how can this be presented for let’s say 1,000 hydrogen atoms? Energy = 1.008 (Hydrogen mass) * 1,000 (hydrogen atoms) * speed of...
  32. K

    I Lie Derivative in Relativity: Examples & Uses

    Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity. I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application...
  33. Math Amateur

    MHB Noetherian Modules - Bland, Example 3, Section 4.2 ....

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to fully understand Example 3, Section 4.2 ... Example 3, Section 4.2 reads as follows: My questions are as...
  34. Math Amateur

    I Noetherian Modules - Bland, Example 3, Section 4.2 ....

    I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to fully understand Example 3, Section 4.2 ... Example 3, Section 4.2 reads as follows: My questions are as...
  35. M

    I Clarification of a specific orbit example

    Hello, In my professor's lecture notes she gives this example and I have a couple of questions regarding it:Let M = \mathbb Z/6\mathbb Z and f: M \rightarrow M, x \rightarrow x+1 the cyclical permutation of the elements from M. Then is G := \{id_M, f, f^2,f^3,f^4,f^5\} a subset from S_M...
  36. Philip Dhingra

    I Tangible example of relativity of simultaneity

    I always liked the idea of starting relativity discussions with the idea that your father could travel at a substantial fraction of the speed of light and return back at roughly the same age as you. This makes time dilation seem interesting, and drives curiosity in students. But I'm struggling...
  37. Math Amateur

    MHB Example on Z-modules .... Dummit & Foote, Page 339 ....

    I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ... I am currently studying Chapter 10: Introduction to Module Theory ... ... I need some help with an aspect of Dummit and Foote's example on Z-modules in Section 10.1 Basic Definitions and Examples ... ... Dummit and...
  38. Math Amateur

    I Example on Z-modules .... Dummit & Foote, Page 339 ....

    I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ... I am currently studying Chapter 10: Introduction to Module Theory ... ... I need some help with an aspect of Dummit and Foote's example on Z-modules in Section 10.1 Basic Definitions and Examples ... ...Dummit and...
  39. Math Amateur

    MHB (R/I)-Modules .... Dummit and Foote Example (5), Section 10.1 ....

    I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ... I am currently studying Chapter 10: Introduction to Module Theory ... ... I need some help with an aspect of Example (5) of Section 10.1 Basic Definitions and Examples ... ... Example (5) reads as follows: I do...
  40. Math Amateur

    I (R/I)-Modules .... Dummit and Foote Example (5), Section 10.1

    I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ... I am currently studying Chapter 10: Introduction to Module Theory ... ... I need some help with an aspect of Example (5) of Section 10.1 Basic Definitions and Examples ... ... Example (5) reads as follows: I...
  41. opus

    Trig Identities -- example problem confusion

    Homework Statement Back with more trig identities. Verify that the following is an identity ##-tan\frac{a}{2} = cot\left(a\right)-csc\left(a\right)## Homework Equations All pythagorean identities, double angle formulas, half angle formulas The Attempt at a Solution In the picture that I've...
  42. M

    MHB I would like to know for example how I prove or disprove this sentence ?

    Hi, I am new here , I have stuck to Proved or disproved This sentence : If for any A,B,C sets exist A\C=B\C And A ∩ C =B ∩ C , then A⊆ B. Thanks very much for help.
  43. C

    MHB Example of systems of the differential linear equations

    Dear Everybody, I have a question about an example: "Solve the system: $x'(t)=3x(t)-4y(t)+1$ $y'(t)=4x(t)-7y(t)+10t$ We write the system using the operator notation: $(D-3)[x]+4y=1$ $-4x+(D+7)[y]=10t$ We can eliminate x from this system by adding 4 times the first equation to $(D-3)$...
  44. G

    A Question About Elementary Fiber Bundle Example

    My question regards the classic example of trivial vs non-trivial fiber bundles, the 'cylinder' and the 'Mobius band.' I'm using Nakahara's 'Geometry, Topology, and Physics', specifically Example 9.1. Here's the text: Then he says (and this is the bit where he loses me): My questions are: i)...
  45. Math Amateur

    MHB Total Derivatives and Linear Mappings .... D&K Example 2.2.5 ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...
  46. Math Amateur

    I Total Derivatives and Linear Mappings .... D&K Example 2.2.5

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 2: Differentiation ... ... I need help with an aspect of Example 2.2.5 ... ... Duistermaat and Kolk's Example 2.2.5 read as follows: In the above text by D&K we...
  47. M

    I Why do these functions form complete orthogonal systems in the Hilbert space?

    Hi PF! A text states that the following two functions $$ \psi^o_k = \sin(\pi(k-1/2)x)\cosh(\pi(k-1/2)(z+h)): k\in\mathbb{N},\\ \psi^e_k = \cos(\pi kx)\cosh(\pi k(z+h)): k\in\mathbb{N} $$ each form complete orthogonal systems in two mutually orthogonal subspaces, which compose the Hilbert...
  48. Math Amateur

    MHB Differentials in Multivariable Functions .... Kantorovitz: Example 4, page 66

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 4 on page 66 ... Kantorovitz's Example 4 on page 66 reads as follows:In the above example, Kantorovitz...
  49. JTC

    A Example of how a rotation matrix preserves symmetry of PDE

    Good Day I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc. But I am making a lot of progress. There is, however, one issue that continues to elude me. I often read how Lie developed Lie Groups to study symmetries of PDE's May I ask if someone...
  50. Math Amateur

    MHB Linear Mappings are Lipschitz Continuous .... D&K Example 1.8.14 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.8.14 ... ... The start of Duistermaat and Kolk's Example 1.8.14 reads as...
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