OpenLLM-France/wikimedia huggingface dataset:
# Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 8
## Problem 1: Part 8

Consider the curved composite beam shown in Figure 12.
* Figure 12. Composite Beam.
Assume that the beam has been shaped into an arc of a circle. The material of the beam is a transversely isotropic fiber composite material with the fibers running along the length of the beam. The rate constitutive relation of the material is given by
    ${\cfrac {D}{Dt}}{\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{31}\\\sigma _{12}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\C_{12}&C_{11}&C_{13}&0&0&0\\C_{13}&C_{13}&C_{33}&0&0&0\\0&0&0&C_{44}&0&0\\0&0&0&0&C_{44}&0\\0&0&0&0&0&(C_{11}-C_{12})/2\end{bmatrix}}{\begin{bmatrix}D_{11}\\D_{22}\\D_{33}\\D_{23}\\D_{31}\\D_{12}\end{bmatrix}}~.$
The problem becomes easier to solve if we consider numerical values of the parameters. Let the local nodes numbers of element $5$ be $1$ for node $5$, and $2$ for node $6$.
Let us assume that the beam is divided into six equal sectors. Then,
    $\theta ={\cfrac {\pi }{4}}~;~~\theta _{1}={\cfrac {\pi }{3}}~;~~\theta _{2}={\cfrac {\pi }{6}}~.$
Let $r_{1}=1$ and $r_{2}=1.2$. Since the blue point is midway between the two, $r=1.1$.
Also, let the components of the stiffness matrix of the composite be
    $C_{11}=10;~~C_{33}=100;~~C_{12}=6;~~C_{13}=60;~~C_{44}=30~.$
Let the velocities for nodes $1$ and $2$ of the element be
    $\mathbf {v} _{1}={\begin{bmatrix}v_{1}^{x}\\v_{1}^{y}\\\omega _{1}\end{bmatrix}}={\begin{bmatrix}1\\2\\1\end{bmatrix}}~;~~\mathbf {v} _{2}={\begin{bmatrix}v_{2}^{x}\\v_{2}^{y}\\\omega _{2}\end{bmatrix}}={\begin{bmatrix}2\\1\\1\end{bmatrix}}$
The Maple code for these initializations is shown below
# Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 4
## Problem 1: Part 4

Why is the Euler-Bernoulli beam theory $C^{1}$? Why is the Timoshenko beam theory $C^{0}$?
In the Euler-Bernoulli theory, the rate of deformation in the $xx$ direction is given by
    $D_{xx}={\frac {\partial v_{x}^{M}}{\partial x}}-y{\frac {\partial ^{2}v_{y}^{M}}{\partial x^{2}}}~.$
Since second-derivatives of $v_{y}^{M}$ are involved, any approximate solution needs to be at least twice differentiable (or $C^{1}$) to be acceptable. This is why this theory is called $C^{1}$.
On the other hand, in the Timoshenko theory, only first derivatives of the primary dependent variables appear. Therefore, once differentiable (or $C^{0}$) shape functions are acceptable and the theory is called $C^{0}$.
# Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 2
## Problem 1: Part 2

Why is the continuum-based approach preferred in nonlinear finite element analysis?
The continuum-based approach is preferred because the governing equations for nonlinear beams and shells are very complex and features such as variations in thickness, junctions, and stiffeners are difficult to incorporate. It is also easier to model complex constitutive models (plasticity, anisotropy) with the continuum-based approach.
Also, codes that use this approach are generally quite robust and accurate.
# Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 5
#### Part 5

Why is a $C^{0}$ beam theory preferable?
A $C^{1}$ approximation is difficult to construct in multi-dimensions (for example, for beams that can deform in the $z$-direction too). That is why $C^{0}$ theories are preferred.
# Nonlinear finite elements/Homework 6/Solutions/Problem 1/Part 6
## Problem 1: Part 6

What are the major assumptions in the Timoshenko beam formulation described in the book chapter (the authors call these the modified Mindlin-Reissner assumptions)?
The assumptions are:
* The fibers (or directors) remain straight.
* The transverse normal stress is negligible (this is also called the plane stress condition.)
* The fibers are inextensible.
# PlanetPhysics/Abel Laureates List
## Abel Award Laureates

The Abel award is an International Award by The Norwegian Academy of Science and Letters which is awarded for outstanding, exceptional work by living mathematicians, mathematical physicists or theoretical physicists. The Niels Henrik Abel (1802--1829) Memorial Fund was established at the bicentenary of Abel's birth (in 2002) to award the Abel Prize for outstanding scientific work in the field of mathematics. The choice of an Abel Laureate is based upon the recommendation by the Abel Committee consisting of five internationally recognized mathematicians, and is handed over by the King of Norway in an official ceremony. Both Jean-Pierre Serre and Mikhail L. Gromov attended the Abel ceremony in May 2009. Norway is reported to have provided in 2002 an initial, total funding of NOK 200,000,000 (over 23 million USD) = in an attempt to generate publicity for mathematics, and increase its popularity especially with young students of mathematics and mathematical physics. One can thus say that to date all of the awards are for advanced mathematics relevant to both `pure' mathematics and physics. ""The Abel Prize recognizes contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of NOK 6,000,000 (close to Euros 700,000, or USD 950,000). Mikhail L. Gromov will receive the Abel Prize from His Majesty King Harald at an award ceremony in Oslo, Norway, May 19. $(27.03.2009)$ .
The following is the current short list of Abel Laureates.
* 2003: Jean-Pierre Serre, ""for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.
* 2004: Sir Michael Francis Atiyah and Isadore M. Singer, ""for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
* 2005: Peter D. Lax, ""for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.
* 2006: Lennart Carleson, ""for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems.
* 2007: Srinivasa S. R. Varadhan, ""for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.
* 2008: John Griggs Thomson (at U. Florida, USA) and Jacques Tits (at Coll\`ege de France, in Paris, France), ""for their profound achievements in algebra and in particular for shaping modern group theory.
* 2009: Mikhail Leonidovich Gromov, received his award for his revolutionary contributions to geometry. Gromov-Witten invariants are of significant interest in modern quantum field theory,the theory of groups of polynomial growth, and also immersion theory. He is a foreign member of the U.S. National Academy of Sciences and of the American Academy of Arts and Sciences, and a Member of l'Acad\'emie Francaise de Sciences (The French Academy of Sciences) (along with Jean-Pierre Serre).
# PlanetPhysics/Abelian Category

The following is the definition of an Abelian category according to Barry Mitchell (1965).

### Definition

An Abelian category is an exact additive category with finite products.
The following theorem from ref. is also relevant as it relates key properties of Abelian categories:
   The following statements are equivalent:
   * ${\mathcal {A}}$ is an Abelian category;
   * ${\mathcal {A}}$ has kernels, cokernels, finite products, finite coproducts, and is both normal and comormal;
   * ${\mathcal {A}}$ has pushouts and pullbacks and is both normal and conormal.
# PlanetPhysics/Affine Connection

An affine connection is a connection defined on the tangent bundle of a manifold. Such connections enjoy many properties which are not true of connections on other bundles. The reason for this goes back to the fact that to define a connection, one needs to consider two vector spaces, the tangent space and the fiber space. In the case of an affine connection, these two spaces are the same, and this leads to possibilities which one would not have when the spaces were distinct.
Perhaps the most important property of affine connections is that they allow one to define geodesics. Because the defining equation of a geodesic involves the covariant derivative of a tangent vector field, only affine connections give rise to geodesics. Furthermore, the converse is true: An affine connection is uniquely determined by the set of its geodesic curves.
Given an affine connection, one can obtain another affine connection by interchanging the role of the vector which specifies direction of parallel transport and the vector being transported. In local coordinates, an expression for this new connection is ${C'}_{jk}^{i}=C_{kj}^{i}$ Since the difference of two connections is a tensor field, one has a tensor field, known as the torsion tensor, associated to every affine connection. $T_{jk}^{i}=C_{jk}^{i}-C_{kj}^{i}$
# PlanetPhysics/Affine Parameter

Given a geodesic curve, an affine parameterization for that curve is a parameterization by a parameter $t$ such that the parametric equations for the curve satisfy the geodesic equation.
Put another way, if one picks a parameterization of a geodesic curve by an arbitrary parameter $s$ and sets $u^{\mu }=dx^{\mu }/ds$, then we have $u^{\mu }\nabla _{\mu }u^{\nu }=f(s)u^{\nu }$ for some function $f$. In general, the right hand side of this equation does not equal zero --- it is only zero in the special case where $t$ is an affine parameter.
The reason for the name "affine parameter" is that, if $t_{1}$ and $t_{2}$ are affine parameters for the same geodesic curve, then they are related by an affine transform, i.e. there exist constants $a$ and $b$ such that
   $t_{1}=at_{2}+b$
Conversely, if $t$ is an affine parameter, then $at+b$ is also an affine parameter.
From this it follows that an affine parameter $t$ is uniquely determined if we specify its value at two points on the geodesic or if we specify both its value and the value of $dx^{\mu }/dt$ at a single point of the geodesic.
# PlanetPhysics/Acceleration

The acceleration of an object is the time derivative of its velocity. Like velocity, acceleration can therefore be considered either as a vector quantity or as a scalar quantity. Acceleration is usually denoted by the symbol $a$, by ${\dot {v}}$ (the time derivative of the velocity) or by ${\ddot {x}}$ (the second time derivative of the position). We can write the definition of acceleration (in vector form) as follows: $\mathbf {a} (t)\equiv {\frac {\mathrm {d} \mathbf {v} (t)}{\mathrm {d} t}}.$
The SI unit of acceleration is $\mathrm {m/s^{2}}$ (metres per second per second, or metres per second squared). Another unit of acceleration is $g$, defined as $g=9.80665\;\mathrm {m/s^{2}}$; this is approximately the acceleration due to gravity at the surface of the Earth at a latitude of $45^{\circ }$.
In addition to acceleration as the time derivative (instantaneous rate of change) of velocity, the average acceleration , or the change of velocity $\Delta \mathbf {v}$ over a specified period of time $\Delta \mathbf {t}$, can also be defined: $\mathbf {\bar {a}} \equiv {\frac {\Delta \mathbf {v} }{\Delta t}}.$
In classical mechanics, acceleration is caused by forces. If a total force $\mathbf {F}$ acts on an object with constant mass $m$, the object undergoes an acceleration $\mathbf {a}$ as described by Newton's second law: $\mathbf {F} =m\mathbf {a} .$ In contrast to velocity, which depends on the observer's system of reference, acceleration can be called an absolute quantity, in the sense that two observers moving with constant velocity with respect to each other perceive the same acceleration.
# PlanetPhysics/Additive Quotient Category 3
### Essential data: Dense subcategory

A full subcategory ${\mathcal {A}}$ of an abelian category ${\mathcal {C}}$ is called dense if for any exact sequence in ${\mathcal {C}}$: $0\to X'\to X\to X''\to 0,$ $X$ is in ${\mathcal {A}}$ if and only if both $X'$ and $X''$ are in ${\mathcal {A}}$.
Remark 0.1: One can readily prove that if $X$ is an object of the dense subcategory ${\mathcal {A}}$ of ${\mathcal {C}}$ as defined above, then any subobject $X_{Q}$, or quotient object of $X$, is also in ${\mathcal {A}}$. \\

#### System of morphisms Σ$_A$

Let ${\mathcal {A}}$ be a dense subcategory (as defined above) of a locally small Abelian category ${\mathcal {C}}$, and let us denote by $\Sigma _{A}$ (or simply only by $\Sigma$ -- when there is no possibility of confusion) the system of all morphisms $s$ of ${\mathcal {C}}$ such that both $kers$ and $cokers$ are in ${\mathcal {A}}$. One can then prove that the category of additive fractions ${\mathcal {C}}_{\Sigma }$ of ${\mathcal {C}}$ relative to $\Sigma$ exists.
The quotient category of Failed to parse (syntax error): {\displaystyle \mathcal{C } relative to ${\mathcal {A}}$}, denoted as ${\mathcal {C}}/{\mathcal {A}}$, is defined as the category of additive fractions ${\mathcal {C}}_{\Sigma }$ relative to a class of morphisms $\Sigma :=\Sigma _{A}$ in ${\mathcal {C}}$. Remark 0.2 In view of the restriction to additive fractions in the above definition, it may be more appropriate to call the above category ${\mathcal {C}}/{\mathcal {A}}$ an additive quotient category . This would be important in order to avoid confusion with the more general notion of quotient category--which is defined as a category of fractions. Note however that Remark 0.1 is also applicable in the context of the more general definition of a quotient category.
# Open Educational Practices/Introduction/Checklist

Open Educational Practices Checklist
This page documents activities completed in the Open Educational Practices course.

## Introduction

* Piazza introduction post
* Link to your Wikiversity user page
* Course wiki page edit
* Piazza reflection posts

## Open Educational Resources

* Piazza 5 Rs post
* Piazza licensing post
* Link to your published textbook review
* Course wiki page edit
* Piazza reflection posts

## Wikis

* Link to your Wikipedia, Wikibooks, or Wikiversity content page edit(s)
* Piazza wiki edit post
* Course wiki page edit
* Piazza reflection posts

## Pressbooks

* Piazza team post(s)
* Link to created Pressbook content
* Piazza feedback post
* Course wiki page edit
* Piazza reflection posts

## Multimedia

* Piazza team post(s)
* Link to created multimedia content
* Piazza feedback post
* Course wiki page edit
* Piazza reflection posts

## Accessibility and UDL

* Piazza team post(s)
* Link to revised content
* Piazza content revision post
* Course wiki page edit
* Piazza reflection posts

## Instructional Design

* Piazza content improvement summary post
* Course wiki page edit
* Piazza reflection posts

## Final Project

* Link to completed lesson plan
* Piazza feedback post
* Course wiki page edit
* Piazza reflection posts
# Open Educational Practices/Introduction/Create Checklist

Enter your username followed by /OEP to create your OEP checklist:
# Open Educational Practices/Introduction/Join

To join the Open Educational Practices course and access Piazza course discussions:
1. Visit https://piazza.com/harpercollege/fall2019/oep .
2. Join as Student and enter OEP as the Class Access Code.
3. Enter your email address.
4. Check your email account and copy the Piazza activation code. Enter the activation code.
5. Set up your Piazza account.
6. (Don't) join the Piazza network.
# Open Educational Practices

Open Educational Practices (OEP) use Open Educational Resources (OER) for teaching and learning in order to innovate the learning process. Open educational practices are represented in teaching techniques that draw upon open technologies and high-quality OER in order to facilitate collaborative and flexible learning.
This course introduces participants to open educational practices. Topics include open educational resources, open content development, accessibility, instructional design, and open pedagogy. Open educational practices are applied throughout the course in a variety of individual and team learning activities.

## Course Outcomes

Participants will:
* Develop an understanding and appreciation for open content and open educational practices
* Identify potential open resources for use in their educational environments
* Create and edit open resources using wikis, Pressbooks, and multimedia
* Apply accessibility and UDL concepts to open resources
* Understand and apply basic instructional design concepts
* Develop a lesson based on open educational practices for use in their classes

## Lessons

* Introduction
* Open Educational Resources
* Wikis
* Pressbooks
* Multimedia
* Accessibility and UDL
* Instructional Design
* Final Project

## See Also

* Open Educational Resources
* Wikipedia: Open educational practices
* OpenPedagogy.org
* BC Open Campus: Open Pedagogy Resources
# PlanetPhysics/Axis Angle of Rotation to Direction Cosine Matrix

For now, without proof you can get the direction cosine matrix (DCM) from the axis angle of rotation by
   $\left[{\begin{matrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{matrix}}\right]=\left[{\begin{matrix}cos(\alpha )+e_{1}^{2}(1-cos(\alpha ))&e_{1}e_{2}(1-cos(\alpha ))+e_{3}sin(\alpha )&e_{1}e_{3}(1-cos(\alpha ))-e_{2}sin(\alpha )\\e_{1}e_{2}(1-cos(\alpha ))-e_{3}sin(\alpha )&cos(\alpha )+e_{2}^{2}(1-cos(\alpha ))&e_{2}e_{3}(1-cos(\alpha ))+e_{1}sin(\alpha )\\e_{1}e_{3}(1-cos(\alpha ))+e_{2}sin(\alpha )&e_{2}e_{3}(1-cos(\alpha ))-e_{1}sin(\alpha )&cos(\alpha )+e_{3}^{2}(1-cos(\alpha ))\end{matrix}}\right]$
# PlanetPhysics/Axis Angle of Rotation to Quaternions

Quaternions can be defined from the axis angle of rotation by
   $\left[{\begin{matrix}q_{1}\\q_{2}\\q_{3}\\q_{4}\end{matrix}}\right]=\left[{\begin{matrix}e_{1}sin(\alpha /2)\\e_{2}sin(\alpha /2)\\e_{3}sin(\alpha /2)\\cos(\alpha /2)\end{matrix}}\right]$
# PlanetPhysics/B Mod Category Equivalence Theorem

\begin{theorem}{\mathbf B-mod category equivalence theorem.}
Let ${\mathcal {A}}$ be an abelian category with arbitrary direct sums (or coproducts). Also, let $P$ in ${\mathcal {A}}$ be a compact projective generator and set $B=(End_{\mathcal {A}}P)^{op}$. The functor $hom_{\mathcal {A}}(P,--)$ yields an equivalence of categories between ${\mathcal {A}}$ and the category $B-mod$. \end{theorem}
Proof. The proof proceeds in two steps. At the first step one shows that the functor $F(X)=hom_{\mathcal {A}}(P,X)$ is fully faithful, and therefore, at the second step one can apply the Abelian category equivalence lemma to yield the sought for equivalence of categories.
# PlanetPhysics/Axioms of Physics

A contributed topic on the Axioms of Physics * in the following areas:
* general relativity Theory
* Axiomatic and Algebraic Quantum Field Theory
* Quantum Algebraic Topology #topological quantum field theory (TQFT) and HQFT**
* quantum logics and Quantum Computing
* Symmetry and Physical Principles and Laws
* categorical physics: category theory and groupoid/ topology applications in Physics
Physical Networks and General dynamics

### Some Fundamental Concepts in Physics

* Space, $s$
* Time, $t$
* Spacetime, $ST$
* reference system or Frame of Reference ($RS$)
* Motion
* Measurement
* observable #Universal constants: Planck's constant ($h$, the speed of light, $c$, gravitational constant, $G$; hyperfine constant, $e$; cosmological constant, $\lambda$
* Matter
* Mass
* Particle
* Elementary Particles and quarks #Energy: Conservation Laws, Hamiltonian operator, energy eigenvalues. eigenstates, Lagrangian, Photons/Electromagnetic radiation #Fields: Electromagnetic, magnetic, electrical; nuclear; gravitational
* Gravity
* String/strings
# PlanetPhysics/Axis Angle of Rotation

The axis angle of rotation is a result of Euler's theorem of rotation and it describes a rotation of a three dimensional rigid body by a unit vector
   ${\hat {e}}=\left[{\begin{matrix}e_{1}\\e_{2}\\e_{3}\end{matrix}}\right]$
and an angle of rotation, $\alpha$.
# PlanetPhysics/Basic Examples of Calculating Work in Physics

Example 1 . Find in the various units (SI, English and CGS) the work done on a mass 112 pounds when lifted through 100 feet.
For this example, the equation for work simplifies to just the force times the distance that the force is acting.
$W=Fd$
Plugging in the numbers for SI units
$W={\frac {112[lb]}{1}}\cdot {\frac {1[slug]}{32.17[lb]}}\cdot {\frac {14.59[kg]}{1[slug]}}\times {\frac {100[ft]}{1}}{\frac {1[m]}{3.281[ft]}}\cdot 9.806[m/s^{2}]=15,180[joules]$
Plugging in the numbers for English units
$W=112[lb]\times 100[ft]=11,200[ft-lb]$
Plugging in the numbers for CGS units
$W={\frac {112[lb]}{1}}\cdot {\frac {453.6[gram]}{1[lb]}}\times {\frac {100[ft]}{1}}\cdot {\frac {1[m]}{3.281[ft]}}\cdot {\frac {100[cm]}{1[m]}}\cdot {\frac {9.806[m/s^{2}]}{1}}\cdot {\frac {100[cm]}{1[m]}}=1.518\times 10^{11}[ergs]$
Example 2 . How much mork is done by a force of $5x$ newtons acting in the x-direction upon a particle while it is displaced from $x=1[m]$ to $x=10[m]$?
The force for this example is a function of $x$
$F(x)=5x$
integrating from position 1 to 2 yields the work
$W=\int _{1}^{10}5xdx={\bigg |}_{1}^{10}{\frac {5}{2}}x^{2}=250-2.5[N\cdot m]=247.5[N\cdot m]$
# Open Web Sources for Teaching Innovation & Entrepreneurship- Links to entrepreneurial websites with useful information

* Links to entrepreneurial websites with useful information
* 1. International Council for Small Business (ICSB)‎
* The International Council for Small Business (ICSB) was the first ‎international membership organization to promote the growth and development of small ‎businesses worldwide. It is devoted to the advancement of management development ‎practices for potential entrepreneurs and existing small business owner/managers through ‎education, research, and the free exchange of ideas.
* The knowledge center offers information on all of ‎ICSB conference proceedings, white papers, innovative teaching resources, and experiential ‎learning materials for registered members. The knowledge links to SME and Entrepenuership Blogs and a number ‎of SME business websites.‎
* ‎2. StartupNation
* StartupNation is a free service founded by entrepreneurs for ‎entrepreneurs. It has grown to be the leading online content and community resource for ‎entrepreneurs. It connects to the local business and entrepreneurial networks in the North ‎America and contains a lot of useful information for budding entrepreneurs. Links to podcasts, ‎articles and business advise blogs can also be found here.‎
* ‎3. Business Link in London - Social Enterprise
* Business Link in London for Social enterprise is a free, impartial, independent business advisory service available to anyone starting up or ‎running a small or medium sized business. BLL provides high quality, actionable information and ‎practical advice, connecting to expert assistance to help one to succeed. BLL website contains ‎excellent free tool kit for setting up social enterprises, guides on business plans to market analysis, ‎videos and many more useful resources for entrepreneurs. The Resource Directory is a valuable repository of ‎case studies, templates , tips and fact sheets on a wide array of entrepreneurial topics. The other ‎BLL links are, [www.businesslink.gov.uk/london/women Women's Online Centre for female entrepreneurs] and [www.businesslink.gov.uk/london/enable ENABLE website ] for deaf and disabled ‎entrepreneurs . ‎
# Open Web Sources for Teaching Innovation & Entrepreneurship- Links to Articles/Publications/Videos/Books

* Articles/Publications/Videos/Books
* ‎1. BBC’s Dragons’ Den
* BBC’s Dragons’Den is a series of video shows on Entrepreneurs ‎pitching for investment in the Den from the ‘ Dragons’, who are five venture capitalists willing to ‎invest their own money into the ventures of the budding entrepreneurs, in exchange for equity. ‎The Dragons Den webpage also contain an online resource bank called the ‘Business School’ to ‎learn about nine essential elements of business. It also connects to many of BBC business links and ‎external sources useful for doing business.‎