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Motivation and emotion/Textbook/Motivation/Self-actualisation

Overview
“Man is a perpetually wanting animal” (Maslow, 1943, p.22)

Self-actualization is commonly understood in light of Maslow's hierarchy of needs. When an individual has satisfied their physiological, safety, love and belongingness and esteem needs then they have reached self actualization. This means the individual is now motivated by a new set of needs which are centred on values such as truth, goodness, beauty, honesty and they all seek to provide greater meaning to the life of the self actualised person. Self actualised individuals readily experience peak performances and they are no longer motivated by deficiencies, instead they are motivated towards growth and reaching their full potential.

What is motivation?
The word motivation comes from the Latin verb movere, which means “to move.” Motivation is thereby concerned with our movements, or actions, and what the determining factors are behind them. More specifically it refers to the processes involved in initiating, maintaining and ceasing goal-oriented behaviours (Maslow, 1970, p. 16). These factors may be internal (which can include drives such as thirst or being hungry or in pain) or external (such as the presence of an attractive person, tasty food or beverages or signs indicating imminent danger). By this definition motivation is the psychological drive that arouses an organism to action toward a desired goal. This concept is also used to explain differences in intensity of behaviour. More intense behaviours are thought to result from higher levels of motivation. Furthermore, motivation can also be used to indicate the persistence of behaviours. For example a highly motivated behaviour will most often be persistent even though the intensity of the behaviour could be low (Franken, 2007 p. 4).

For more information, see What is motivation?

Holism and The Humanistic Perspective in Psychology
Humanism is a school of thought in psychology which developed at the turn of the twentieth century. It emerged as a result of dissatisfaction with psychoanalytic and behaviourist models that were viewed as insufficient or outdated (Schneider, Bugental & Pierson, 2001). Humanistic psychology is characterised by the search for human potential and encouraging development. This same view stressed the role of personal choice and the importance of personal growth and self actualization (Humanistic Psychology, 2009). One of the main proponents of Humanism was Carl Rogers, an influential American psychologist who was one of the most prominent theorists developing the notion of holism. Holism is a key theoretical component of the humanistic school of thought which rejects the reductionist point of view about the human condition. Instead it believes that the properties of an individual cannot be determined or explained by its component parts alone; rather the system as a whole determines in an important way how the parts behave (Pribram, 1979). Within the context of motivation, no need or drive can be treated as if it were discrete or isolated. Every drive is viewed as related to the state of satisfaction or the dissatisfaction of other drives (Harriman, 1970). This is a dominant notion throughout the humanistic perspectives, especially self actualization (Pribram).

Abraham Maslow on Self Actualiztion
"A musician must make music, an artist must paint, a poet must write, if he is to be ultimately happy. What a man can be, he must be. This need we can call self actualization (Maslow, 1943 p. 34)".

, 1908-1970]]

Abraham Maslow (1943) famously elaborated on the term self actualization soon after the term was coined by Kurt Goldstein. Maslow took a more specific and limited lens on self actualization; referring it to a person's desire for fulfilled and the desire to achieve everything within their potential. Maslow followed on to describe self actualization as our tendency to become increasingly closer to who we are and to become actualized in everything we are capable of. Maslow described this term in light of his hierarchical theory on human motivation. He believed that upon achieving all of the preceding lower order needs, a person's desire to become self actualized will take form in varying ways amongst different people (Maslow, 1943). Maslow's definition of self actualization developed through early interviews with people he both knew and admired. He then interviewed another sample of people he thought were self-actualized. He would wrote down a list of common traits he identified among each of the subjects, searching for some correlation. Maslow then took that trait list and analysed a second sample of self-actualized individuals who matched up with many of the the key traits. Famous People which Maslow felt were self-actualized included Thomas Jefferson, Abraham Lincoln, Albert Einstein, Jane Addams, William James, Albert Schweitzer, Aldous Huxley and Elanor Roosevelt. Through a process of refining his trait list again and again, Maslow eventually came up with what he believed to be a stable list of attributes which would define the self-actualised individual (Maslow, 1943). Some of these most notable attributes include being honest and being involved in a cause outside of one's self and experiencing fully, vividly, and selflessly with full concentration and absorption. Self actualized people were also described as spontaneous and as having profound interpersonal relationships, meaning that their serious relationships are few yet deep. Another of the main traits Maslow associated with self actualised individuals was a philosophical sense of humour and high capacity for Creativity (Maslow, 1943).

See also

Description of theory
Maslow’s hierarchy of needs (1943) is a motivational theory that emphasises the humanistic view of striving to reach one's full potential as a basic human motivation. This theory is often represented visually as a pyramid split into five sections. At the bottom are the most fundamental human needs and higher order needs are situated progressively up the pyramid. Maslow believed that human needs arrange themselves in this prepotency. This meant that the appearance of one need is generally reliant on the prior satisfaction of another more basic or pre-potent needs. Upon satisfying all of the lower order needs, a person is said to reach the ultimate pinnacle of this model, self actualization.

Physiological needs
As the name suggests, physiological needs are the literal requirements for human survival. These are the requirements for the human body to simply continue functioning. This primitive level of motivation includes breathing, nutrition and homeostasis. Physiological needs also include metabolic requirements such as water and food as well as clothing and shelter which are necessary to provide protection from the elements. These are thought to be the strongest needs because if a person were deprived of all needs, the physiological ones would come first in the person's search for satisfaction (Maslow, 1943 p.26).

Safety needs
Once an individual’s physiological needs have been fulfilled, humans are then motivated towards satisfying their safety needs. Safety and Security requirements in modern people take the form in the need for personal security, financial security the need for law and social order, health and well being and having a safety net against accidents, illness and their adverse impacts. Adults are thought to have little awareness of their safety/ security needs except in times of emergency or periods of disorganization in the social structure (such as widespread rioting). Conversely children regularly display the signs of insecurity and the need to be safe. Interestingly again, safety needs can never be over satisfied; and according to this hierarchy, if a individual feels threatened, higher order needs will not receive attention until that threat has been resolved (Maslow, 1943 p.28).

Love and belonging
Once physiological and safety needs are satisfied, the third layer of human needs are social and involve feelings of belongingness. This is the first of what Maslow describes as our higher order needs. Love and belonging needs involve all emotionally based relationships such as friendships, family and intimate relationships. These needs can be expressed through an individual’s desire to seek out friendships, belong to a group and to give and receive love. These needs might be expressed through the search for a mate and the desire to be part of a family. In the absence of these elements people can become susceptible to loneliness, social anxiety and depression. Maslow believes that this level is what the majority of the population remains at (Maslow, 1943 p.32).

Esteem
Once the individual has achieved love and belonging the next higher order need is that of esteem. Maslow distinguished between two levels of esteem needs: reputation and Self-Esteem. Esteem needs related to reputation include a person’s desire for attention, recognition and social status. Self esteem needs on the other hand consist of the desire for accomplishment and self respect. When these needs are satisfied, the person develops confidence and feels like a valuable person in the world. When these needs are frustrated, it can lead to an inferiority complex, and the individual can feel weak, helpless and worthless (Maslow, 1943 p.33).

Self actualization
Once all four of these needs have been met, the individual has reached the summit of Maslow’s hierarchy of needs; self actualization. The individual’s needs then become about the quest of reaching their full potential as a person. Self actualised people tend to have motivators such as justice, wisdom, truth and meaning. Maslow describes self actualization as a person’s need to be and do that which the person was born to do. Unlike lower level needs, this need is never fully satisfied. As the person grows psychologically, new opportunities are thought to arise; presenting new ways for the individual to continue to grow (Maslow, 1943 p.34).

behaviours characterised by self actualised people
Self actualised people share a range of characteristics, most of which contribute to the development of their greater potential. By definition Self Actualized People must be free from psychopathology and have progressed through the hierarchy of basic needs. Another essential criterion for becoming self-actualized is to realize your need to grow and develop, and to increasingly strive to become who you are fully capable of becoming. Self actualised individuals are not static beings who embrace the status quo, rather, than embrace change, because change is necessary for growth. Other traits correlated among self actualised individuals innclue listening to their own voice, taking responsibility, being honest and being involved in a cause outside of themselves. They experience fully, vividly, and selflessly with full concentration and absorption (Maslow, 1965). Self actualised individuals are thought to have a more efficient perception of reality and are better able to distinguish between fact and fiction. They live with spontaneity and without artifice and have greater acceptance of themselves, others and nature. Among the traits associated with self actualised people they tended to be more problem-centred, instead of ego-centred, be more independent and autonomous and have a higher need for privacy. One notable characteristic found across nearly al self actualise individuals was the presence of profound interpersonal relationships, meaning that their serious relationships are few yet deep. Self-actualisers also tend to be friendly to people without regard to race, gender, age, ethnicity, or social status and are resistant to enculturisation. Another of the main traits associated with self actualised individuals was a philosophical sense of humour and high capacity for creativity. Furthermore, self actualised people are more likely to experience B-Love (love for the essence or being of the other). Maslow makes the distinction between this type of love and D-Love (deficiency love) (in which someone loves another person solely because they are driven to satisfy their needs for love and belongingness) (Maslow, 1943; Showstrom, 1964).

Further research on self actualised individuals revealed that they are more successful in initiating, developing, and maintaining effective, fulfilling relationships with other people (Johnson, 1972 p.14). Self actualised individuals are also thought to be correlated with higher levels of emotional intelligence, better outcomes at work, better general health and well being (Ciarrochi, Forgas & Mayer, 2001 p. 83-90), and better mental health (Knapp, 1965).

What don’t we know
"Since, in our society, basically satisfied people are the exception, we do not know much about self actualization, either experimentally or clinically. It remains a challenging problem for research (Maslow, 1943 p. 35)."

Measuring Self Actualization
One area of self actualization that requires further understanding is how to accurately measure this construct. Early attempts at quantifying a person's degree of self actualization included Showstrom’s Personality Orientation Inventory developed in 1963. This consisted of 150 two choice comparative values aimed at quantifying an individual’s levels of self actualization (Showstrom, 1964). An example of this is choosing: (a) two people will get along best if each concentrates on pleasing the other person; or (b) two people will get along best if each person feels free to express themselves. Within Showstrom’s inventory there are 2 major scales, and 10 subscales. The first of the major scales measures "present orientation", which represents how much an individual embraces as existential approach to life. The second major scale measures "self vs other" orientation, or Ego-centrism. The ten subscales then examine the individual character traits of self-actualized individuals as described by Maslow (Showstrom).

The second notable attempt at measuring self-actualizing tendency was the Short Index of Self-Actualization (Jones & Crandell, 1986). This is a 15-item index of self-actualization developed primarily on modified items from Showstrom's Personal Orientation Inventory. In fact the Short Index uses 15 items from the POI to which the participant must state their agreement on a 6 point likert scale (Strongly Disagree - Strongly Agree). Jones and Crandell claim this made the test much easier to administer and grade than the POI, and also reduces irritability due to the forced choice format of the Personality Orientation Inventory. Despite these improvements the tests creators still recognised a range of weaknesses associated with their scale and say these must one day be overcome in a bid to gain a truly accurate reflection of an individual's self actualising tendencies (Jones & Crandell).

Critiques of Maslow’s hierarchy and self actualization
Maslow's hierarchy of needs is the most widely accepted theory of human motivation. Despite this there a still a range of criticisms directed at the theory. Firstly, critics claim that Maslow was not practising a credible, rigorous scientific methodology with his study. Some researchers feel that Maslow's work relied too heavily on case studies and that not enough experimental work was done on self actualization. One example of Maslow conducting some real research was an early experiment where he screened 3000 university students in 1935. When he came up with only one suitable subject who met the criteria for self actualization, he abandoned the study (Ciarrochi, Forgas & Mayer, 2001 p. 89).

Further criticism is directed at the fact that Maslow’s recognition of self-actualized individuals was almost exclusively limited to Highly Educated White Males. Critics believe that analysis of personality based solely upon the upper stratum of the dominant culture has failed to deliver a truly universal description of personality. Some of these same critics also claim that implicit sexism, racism, and classism stem from Maslow’s work and therefore does not represent a valid way of understanding basic human personality (Nevis, 1983).

Critics then turn to specific examples of Maslow's hierarchy and how it can fail to account for some motives. One such example is people suffering from eating disorders, whereby a need for belonging can often overcome the physiological and security needs, depending on the strength of the peer pressure. An anorexic, for example, may completely ignore the need to eat and the security of health for a feeling of control and belonging (Beck, 2004 p.121). Another contemporary example is that of suicide bombers. These individuals are seemingly willing to ignore all of the most basic principles in Maslow’s hierarchy in order to sacrifice themselves for a principle or cause. In this instance the individual’s hierarchy of motives does not match that represented by Maslow (Baum, 2009).

Within literature on self actualization it is also important to note that even Maslow admits the self actualised individuals are not perfect. He pointed out that although they may not have many of the lesser failings common to all of us. Self actualised people can still be silly, thoughtless and wasteful. They can also be boring, irritating and stubborn and may exhibit a superficial vanity concerning their own products and can show ruthlessness. Self actualised individuals may also feel guilt, sadness, anxiety and conflict arising from their realisation that they have not reached their full potential (Petri & Govern, 2004 p. 351).

Maslow Hierarchy of needs and ERG theory of motivational forces in organisations
Maslow’s hierarchy seems to also hold true in industrial situations. An example of this is lower level workers who seem to be more motivated by money (which is required for food and shelter) and often lack any motivation to be creative in their jobs. Workers at higher levels generally have income sufficient to satisfy many of their lower order needs, and in this instance self actualization is often more important (Beck, 2004 p. 401).

In a bid to address some of the limitations of Maslow's hierarchy, Clayton Alderfer proposed the ERG theory of motivation, which describes our needs in a different hierarchy. The letters ERG stand for three levels of needs: Existence, Relatedness, and Growth. These can be compared to Maslow's hierarchy in that existence represents physiological and safety needs, relatedness accounts for social and external esteem needs and growth encompasses self-actualization and internal esteem needs (Alderfer, 1969).

ERG theory differs from Maslow's hierarchy in that it allows the order of the needs be different for different people and it allows for different levels of needs to be pursued simultaneously. In further contrast ERG theory states that if a higher level need remains unfulfilled, the person may regress to a lower level need that appears easier to satisfy. This phenomenon called the frustration-regression principle (Alderfer, 1969; Alderfer & Guzzo, 1979).

ERG theory also lends itself well to organisational application and it helps managers to recognise that an employee has multiple needs that may need to be satisfied simultaneously. Application of ERG theory in the workplace facilitates managers in understanding how pay, relatedness with group and supervisors, and growth in one's job impact on the motivations of different employees (Alderfer & Guzzo, 1979). This allows for more effective management which can mean tailoring organisational conditions to maximise worker satisfaction and productivity and also allows managers to predict employee reactions to different organisational conditions (Alderfer & Guzzo; Chang & Yuan, 2008). {{Hide in print|

Learning Quiz
{The term "self actualization" was first coined by, + Kurt Goldstein - Abraham Maslow - Adrian Maslow
 * type=""}

{ Within Maslow's Hierarchy of needs food is classified as a { physiological } need whilst recognition is an { esteem } need.
 * type="{}"}

{ This term is used to describe the processes involved in initiating, maintaining and ceasing goal-oriented behaviour: { motivation }
 * type="{}"}

{The 3 letters in Clayton Alderfers ERG theory stands for existence, relatedness and: - games - grieving + growth
 * type=""}

{ Self actualization has been correlated with higher levels of emotional { intelligence }.
 * type="{}"}


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User:Jiangpx/6321 hw6

=Problem 2 - Non-homogeneous L2-ODE-VC=

Given
$$\begin{align} & (1)\cdots xy''+2y'+xy=\sin (x) \\ & (2)\cdots (x-1)y''-xy'+y=\sin (x) \\ \end{align}$$

Find
The complete Solutions

Solution
(1)

We know the homogeneous solutions are

$$\begin{align} & {{u}_{1}}=\frac{\cos x}{x} \\ & {{u}_{2}}=\frac{\sin x}{x} \\ \end{align}$$

The particular solution is given as

$${{y}_{p}}={{u}_{1}}(x)\int{\frac{1}{h(x)}\left( \int{h(x)\frac{f(x)}{{{u}_{1}}(x)}dx} \right)}dx$$

where

$$\begin{align} & h(x)={{u}_{1}}^{2}(x)\cdot \exp \left( \int{{{a}_{1}}(x)dx} \right) \\ & {{a}_{1}}(x)=\frac{2}{x} \\ \end{align}$$

Then we may get h(x) as

$$h(x)=\frac{{{\cos }^{2}}x}\cdot \exp \left( \int{\frac{2}{x}dx} \right)={{\cos }^{2}}x$$

Then the particular solution can be evaluated as follows

$$\begin{align} & {{y}_{p}}=\frac{\cos x}{x}\cdot \int{\frac{1}{{{\cos }^{2}}x}\left( \int{{{\cos }^{2}}x\frac{\sin x/x}{\cos x/x}dx} \right)}dx \\ & {{y}_{p}}=\frac{\cos x}{x}\cdot \int{\frac{1}{{{\cos }^{2}}x}\left( \int{(\cos x)(\sin x)dx} \right)}dx \\ & {{y}_{p}}=\frac{\cos x}{x}\cdot \int{\frac{1}{{{\cos }^{2}}x}\left( \frac{-1}{2}{{\cos }^{2}}x \right)}dx \\ & {{y}_{p}}=\frac{\cos x}{x}\cdot \frac{-x}{2} \\ & {{y}_{p}}=-\frac{1}{2}\cos x \\ \end{align}$$

Therefore, the complete solution can be written as

$$y(x)=\frac{{{k}_{1}}\cos x}{x}+\frac{{{k}_{2}}\sin x}{x}-\frac{1}{2}\cos x$$

where k1 and k2 are integration constants.

(2)

Two homogeneneous solutions are

$$\begin{align} & {{u}_{1}}=\exp (x) \\ & {{u}_{2}}=x \\ \end{align}$$

With

$${{a}_{1}}(x)=\frac{-x}{x-1}$$

Integrating factor h(x) is

$$\begin{align} & h(x)={{[\exp (x)]}^{2}}\cdot \exp \left( \int{\frac{-x}{x-1}dx} \right) \\ & h(x)=\frac{\exp (x)}{x-1} \\ \end{align}$$

Then particular solution is

$$\begin{align} & {{y}_{p}}={{u}_{1}}(x)\int{\frac{1}{h(x)}\left( \int{h(x)\frac{f(x)}{{{u}_{1}}(x)}dx} \right)}dx \\ & {{y}_{p}}=\exp (x)\cdot \int{\frac{x-1}{\exp (x)}}\left( \int{\frac{\exp (x)}{x-1}\cdot \frac{\sin x/(x-1)}{\exp (x)}dx} \right)dx \\ & {{y}_{p}}=\exp (x)\cdot \int{\frac{x-1}{\exp (x)}\left( \int{\frac{\sin x}dx} \right)dx} \\ & \\ \end{align}$$

Therefore, the complete solution can be written as

$$y(x)={{k}_{1}}\exp (x)+{{k}_{2}}x+\exp (x)\cdot \int{\frac{x-1}{\exp (x)}\left( \int{\frac{\sin x}dx} \right)dx}$$

=Problem 3 - Solve non-homogeneous L2-ODE-VC using direct method=

Given
$$\begin{align} & (1-{{x}^{2}}){y}''-2x{y}'+2y=\frac{1}{1-{{x}^{2}}} \\ & {{u}_{1}}(x)=x \\ \end{align}$$

Find
The complete solution $$y(x)$$ using the direct method

Solution
Method 1 - Wronskian

From previous homework, we obtained the two homogeneous solution as

$$\begin{align} & {{u}_{1}}(x)=x \\ & {{u}_{2}}(x)=\frac{x}{2}\log (\frac{1+x}{1-x})-1 \\ \end{align}$$

The complementary function is

$${{y}_{h}}=A{{u}_{1}}(x)+B{{u}_{2}}(x)$$

The particular solution can be obtained via the method of variation of parameters

$${{y}_{p}}=\int\limits_{x}{\frac{{{u}_{1}}(s){{u}_{2}}(x)-{{u}_{1}}(x){{u}_{2}}(s)}{(1-{{s}^{2}})({{u}_{1}}(s){{u}_{2}}'(s)-{{u}_{1}}'(s){{u}_{2}}(s))}ds}$$

Using Abel's formula

$$W(x)=W({{x}_{0}})\exp \left\{ -\int_^{x}{{{a}_{1}}(s)ds} \right\}$$

With

$${{a}_{1}}(s)=\frac{-2s}{1-{{s}^{2}}}$$ the Wronskian can be expressed as

$$W=({{u}_{1}}(s){{u}_{2}}'(s)-{{u}_{1}}'(s){{u}_{2}}(s))=\frac{1-{{s}^{2}}}$$

With

$$W(0)=1$$

The particular solution is

$$\begin{align} & {{y}_{p}}(x)={{u}_{2}}(x)\int{{{u}_{1}}(x)dx}-{{u}_{1}}(x)\int{{{u}_{2}}(x)dx} \\ & {{y}_{p}}(x)=\frac{2}{{u}_{2}}(x)-{{u}_{1}}(x)\left\{ \frac{{{x}^{2}}-1}{4}\log (\frac{1+x}{1-x})-\frac{x}{2} \right\} \\ & {{y}_{p}}(x)=\frac{2}\left[ \frac{x}{2}\log (\frac{1+x}{1-x})-1 \right]-x\left\{ \frac{{{x}^{2}}-1}{4}\log (\frac{1+x}{1-x})-\frac{x}{2} \right\} \\ & {{y}_{p}}(x)=\frac{x}{4}\log (\frac{1+x}{1-x}) \\ \end{align}$$

Method 2 - Direct Method

1st homogeneous solutions is

$${{u}_{1}}(x)=x$$

With $${{a}_{1}}(x)=\frac{-2x}{1-{{x}^{2}}}$$

The integrating factor is

$$\begin{align} & h(x)={{u}_{1}}^{2}(x)\cdot \exp \left( \int{{{a}_{1}}(x)dx} \right) \\ & h(x)={{x}^{2}}\cdot \exp \left( \int{\frac{-2x}{1-{{x}^{2}}}dx} \right) \\ & h(x)={{x}^{2}}\cdot \exp \left( \log (1-{{x}^{2}}) \right) \\ & h(x)={{x}^{2}}(1-{{x}^{2}}) \\ \end{align}$$

2nd homogeneous solution can be obtained as

$${{u}_{2}}(x)={{u}_{1}}(x)\int{\frac{dx}{h(x)}}$$

Then

$${{u}_{2}}(x)=x\int{\frac{dx}{{{x}^{2}}(1-{{x}^{2}})}}=x\left( \frac{1}{2}(\log \frac{1+x}{1-x}-\frac{2}{x}) \right)=\frac{x}{2}\log \frac{1+x}{1-x}-1$$

The particular solution can be obtained as

$${{y}_{p}}={{u}_{1}}(x)\int{\frac{1}{h(x)}}\left( \int{h(x)\frac{f(x)}{{{u}_{1}}(x)}dx} \right)dx$$

with $$f(x)=\frac{1}$$

Then

$$\begin{align} & {{y}_{p}}=x\int{\frac{1}{{{x}^{2}}(1-{{x}^{2}})}}\left( \int{{{x}^{2}}\cdot (1-{{x}^{2}})\frac{1}{{{(1-{{x}^{2}})}^{2}}\cdot x}dx} \right)dx \\ & \cdots =x\int{\frac{1}{{{x}^{2}}(1-{{x}^{2}})}}\left( \int{\frac{xdx}{1-{{x}^{2}}}} \right)dx \\ & \cdots =-\frac{x}{2}\int{\frac{\log (1-{{x}^{2}})}{{{x}^{2}}(1-{{x}^{2}})}}dx \\ & \\ \end{align}$$

Therefore, the complete solution can be written as

$$\begin{align} & y(x)={{k}_{1}}{{u}_{1}}+{{k}_{2}}{{u}_{2}}+{{y}_{p}} \\ & \cdots \cdots ={{k}_{1}}\cdot (x)+{{k}_{2}}\cdot \left( \frac{x}{2}\log \left( \frac{1+x}{1-x} \right)-1 \right)-\frac{x}{2}\int{\frac{\log (1-{{x}^{2}})}{{{x}^{2}}(1-{{x}^{2}})}}dx \\ \end{align}$$

Given
Spherical Coordinates

$$\begin{align} & x=r\cos \theta \cos \phi \\ & y=r\cos \theta \sin \phi \\ & z=r\sin \theta \\ \end{align}$$

Find
The length of an infinitesimal $$d{{s}^{2}}$$ and the Laplacian $$\Delta \psi $$

Solution
Just as in the cartesian coordinate system,

$$d{{s}^{2}}=d{{x}^{2}}+d{{y}^{2}}+d{{z}^{2}}$$

In spherical coordinate system, think of a little cube with three perpendicular sides

$$\langle dr\cdots rd\theta \cdots r\cos \theta d\phi \rangle $$

then

$$\begin{align} & d{{s}^{2}}={{(dr)}^{2}}+{{(rd\theta )}^{2}}+{{[(r\cos \theta )d\phi ]}^{2}} \\ & d{{s}^{2}}=d{{r}^{2}}+{{r}^{2}}d{{\theta }^{2}}+{{r}^{2}}{{\cos }^{2}}\theta d{{\phi }^{2}} \\ \end{align}$$

The Laplacian is given as $$\Delta \psi =\frac{1}\sum\limits_{i=1}^{3}{\frac{\partial }{\partial {{\xi }_{i}}}\left[ \frac\frac{\partial \psi }{\partial {{\xi }_{i}}} \right]}$$

with $$\begin{align} & {{h}_{1}}=1 \\ & {{h}_{2}}=r \\ & {{h}_{3}}=r\cos \theta \\ \end{align}$$

Therefore,

$$\begin{align} & \Delta \psi =\frac{1}{{{r}^{2}}\cos \theta }\left\{ \frac{\partial }{\partial r}\left[ \frac{{{r}^{2}}\cos \theta }\frac{\partial \psi }{\partial r} \right]+\frac{\partial }{\partial \theta }\left[ \frac{{{r}^{2}}\cos \theta }\frac{\partial \psi }{\partial \theta } \right]+\frac{\partial }{\partial \phi }\left[ \frac{{{r}^{2}}\cos \theta }\frac{\partial \psi }{\partial \phi } \right] \right\} \\ & \Delta \psi =\frac{1}\frac{\partial }{\partial r}\left[ {{r}^{2}}\frac{\partial \psi }{\partial r} \right]+\frac{1}{{{r}^{2}}\cos \theta }\left[ \frac{\partial }{\partial \theta }\left( \cos \theta \frac{\partial \psi }{\partial \theta } \right) \right]+\frac{1}{{{r}^{2}}{{\cos }^{2}}\theta }\left[ \frac{{{\partial }^{2}}\psi }{\partial {{\phi }^{2}}} \right] \\ \end{align}$$

=Problem 5 - Laplacian in Cylindrical Coordinate System=

Given
$$\begin{align} & {{x}_{1}}=x=r\cos \phi ={{\xi }_{1}}\cos {{\xi }_{2}} \\ & {{x}_{2}}=y=r\sin \phi ={{\xi }_{1}}\sin {{\xi }_{2}} \\ & {{x}_{3}}=z={{\xi }_{3}} \\ \end{align}$$

Find
$$\begin{align} & (1)Express\left\{ d{{x}_{i}} \right\}\operatorname{in}\cdot terms\cdot of\cdot \left\{ d{{\xi }_{i}} \right\}and\left\{ {{\xi }_{i}} \right\} \\ & (2)Find\cdot d{{s}^{2}}=\sum\limits_{i=1}^{3}{dx_{i}^{2}}=\sum\limits_{i=1}^{3}{h_{i}^{2}\cdot d\xi _{i}^{2}}\cdot and\cdot Express\left\{ {{h}_{i}} \right\}in\cdot terms\cdot of\cdot \left\{ {{\xi }_{i}} \right\} \\ & (3)Find\cdot Laplacian\cdot \Delta \psi \\ & (4)Obtain\cdot Bessel\cdot Equation\cdot by\cdot solving\cdot \Delta \psi =0 \\ \end{align}$$

Solution
(1)

$$\begin{align} & dx=\frac{\partial x}{\partial r}dr+\frac{\partial x}{\partial \phi }d\phi =\cos \phi \cdot dr+(-r\sin \phi )\cdot d\phi \\ & dy=\frac{\partial y}{\partial r}dr+\frac{\partial y}{\partial \phi }d\phi =\sin \phi \cdot dr+(r\cos \phi )\cdot d\phi \\ & dz=dz \\ \end{align}$$

or

$$\begin{align} & d{{x}_{1}}=\cos {{\xi }_{2}}\cdot d{{\xi }_{1}}-{{\xi }_{1}}\sin {{\xi }_{2}}\cdot d{{\xi }_{2}} \\ & d{{x}_{2}}=\sin {{\xi }_{2}}\cdot d{{\xi }_{1}}+{{\xi }_{1}}\cos {{\xi }_{2}}\cdot d{{\xi }_{2}} \\ & d{{x}_{3}}=d{{\xi }_{3}} \\ \end{align}$$

(2)

$$\begin{align} & d{{s}^{2}}={{(dr)}^{2}}+{{(rd\phi )}^{2}}+{{(dz)}^{2}} \\ & d{{s}^{2}}=d{{r}^{2}}+{{r}^{2}}d{{\phi }^{2}}+d{{z}^{2}} \\ & d{{s}^{2}}=d\xi _{1}^{2}+\xi _{1}^{2}d\xi _{2}^{2}+d\xi _{3}^{2} \\ \end{align}$$

with

$$\begin{align} & {{h}_{1}}=1 \\ & {{h}_{2}}=r={{\xi }_{1}} \\ & {{h}_{3}}=1 \\ \end{align}$$

(3)

The Laplacian is given as

$$\Delta \psi =\frac{1}\sum\limits_{i=1}^{3}{\frac{\partial }{\partial {{\xi }_{i}}}\left[ \frac\frac{\partial \psi }{\partial {{\xi }_{i}}} \right]}$$

With $${{h}_{1}}{{h}_{2}}{{h}_{3}}={{\xi }_{1}}$$, The Laplacian becomes

$$\begin{align} & \Delta \psi =\frac{1}\left\{ \frac{\partial }{\partial {{\xi }_{1}}}\left( {{\xi }_{1}}\frac{\partial \psi }{\partial {{\xi }_{1}}} \right)+\frac{\partial }{\partial {{\xi }_{2}}}\left( \frac{1}\frac{\partial \psi }{\partial {{\xi }_{2}}} \right)+\frac{\partial }{\partial {{\xi }_{3}}}\left( {{\xi }_{1}}\frac{\partial \psi }{\partial {{\xi }_{3}}} \right) \right\} \\ & \Delta \psi =\frac{1}\frac{\partial }{\partial {{\xi }_{1}}}\left( {{\xi }_{1}}\frac{\partial \psi }{\partial {{\xi }_{1}}} \right)+\frac{1}{\xi _{1}^{2}}\frac{{{\partial }^{2}}\psi }{\partial \xi _{2}^{2}}+\frac{{{\partial }^{2}}\psi }{\partial \xi _{3}^{2}} \\ \end{align}$$

(4)

Set the Laplacian to zero and write it as follows

$$\Delta \psi =\frac{1}{r}\frac{\partial }{\partial r}\left( r\frac{\partial \psi }{\partial r} \right)+\frac{1}\frac{{{\partial }^{2}}\psi }{\partial {{\phi }^{2}}}+\frac{{{\partial }^{2}}\psi }{\partial {{z}^{2}}}$$

Assume

$$\psi (r,\phi ,z)=R(r)\cdot Phi(\phi )\cdot Z(z)$$

Then

$$\Delta \psi =Phi\cdot Z\cdot \frac{1}{r}\frac{d}{dr}\left( r\frac{dR}{dr} \right)+R\cdot Z\cdot \frac{1}\frac{{{d}^{2}}Phi}{d{{\phi }^{2}}}+R\cdot Phi\cdot \frac{{{d}^{2}}Z}{d{{z}^{2}}}=0$$

Divide by $$R\cdot Phi\cdot Z$$, we may get

$$\frac{1}{R\cdot r}\frac{d}{dr}\left( r\frac{dR}{dr} \right)+\frac{1}{Phi\cdot {{r}^{2}}}\frac{{{d}^{2}}Phi}{d{{\phi }^{2}}}+\frac{1}{Z}\frac{{{d}^{2}}Z}{d{{z}^{2}}}=0$$

Assume

$$\frac{1}{Z}\frac{{{d}^{2}}Z}{d{{z}^{2}}}=k$$

Multiply by $${{r}^{2}}$$ we may get

$$\frac{r}{R}\frac{d}{dr}\left( r\frac{dR}{dr} \right)+\frac{1}{Phi}\frac{{{d}^{2}}Phi}{d{{\phi }^{2}}}+{{(rk)}^{2}}=0$$

Define

$$\frac{1}{Phi}\frac{{{d}^{2}}Phi}{d{{\phi }^{2}}}=-{{m}^{2}}$$

Then

$$\begin{align} & r\frac{d}{dr}\left( r\frac{dR}{dr} \right)+\left( {{(rk)}^{2}}-{{m}^{2}} \right)R=0 \\ & {{r}^{2}}\frac{{{d}^{2}}R}{d{{r}^{2}}}+r\frac{dR}{dr}+\left( {{(rk)}^{2}}-{{m}^{2}} \right)R=0 \\ \end{align}$$

Define $$x=rk$$, then

$$\begin{align} & \frac{dR}{dr}=\frac{dR}{dx}\frac{dx}{dr}=k\frac{dR}{dx} \\ & \frac{{{d}^{2}}R}{d{{r}^{2}}}=\frac{d}{dr}\left( k\frac{dR}{dx} \right)=k\frac{d}{dx}\left( k\frac{dR}{dx} \right)={{k}^{2}}\frac{{{d}^{2}}R}{d{{x}^{2}}} \\ \end{align}$$

Therefore

$$\begin{align} & {{r}^{2}}{{k}^{2}}\frac{{{d}^{2}}R}{d{{x}^{2}}}+rk\frac{dR}{dx}+\left( {{(rk)}^{2}}-{{m}^{2}} \right)R=0 \\ or \\ & {{x}^{2}}\frac{{{d}^{2}}R}{d{{x}^{2}}}+x\frac{dR}{dx}+({{x}^{2}}-{{m}^{2}})R=0 \\ \end{align}$$

Thus Bessel's equation is obtained.

=Problem 6 - Lapalcian in Spherical Coordinate System with Math/Phy Convention=

Given
Solution of Problem 4 and definition of $$\theta $$ measured clockwisely from the North Pole

Find
The expression of Lapalcian

Solution
$$\Delta \psi =\frac{1}\frac{\partial }{\partial r}\left[ {{r}^{2}}\frac{\partial \psi }{\partial r} \right]+\frac{1}{{{r}^{2}}\sin \theta }\left[ \frac{\partial }{\partial \theta }\left( \sin \theta \frac{\partial \psi }{\partial \theta } \right) \right]+\frac{1}{{{r}^{2}}{{\sin }^{2}}\theta }\left[ \frac{{{\partial }^{2}}\psi }{\partial {{\phi }^{2}}} \right]$$

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User:Jiangpx/6321 hw6


text:
Talk:Motivation and emotion/Textbook/Motivation and personality

Feedback
Here's some feedback on your draft so far - hope this helps. -- Jtneill - Talk - c 05:01, 5 November 2010 (UTC)
 * 1) Introduction
 * 2) I tweaked the introduction a bit - see what you think and of course keep rewriting it as you see fit. I really like that you've made a concerted effort to connect personality with motivation.
 * 3) Check APA citation style throughout - there are many small proofreading corrections to be made for good APA style for citations
 * 4) Headings
 * 5) Add a next level of heading using ==== instead of the underlines - this way these headings will appear in the tables of contents (TOC)
 * 6) 5-factor personality model
 * 7) Rewrite: "Another study that researched the relationship between the five personality traits and the motivation to learn found extraversion, in particular the facets of activity and positive emotions, openness to experience, in particular the facets of ideas and values and conscientiousness, in particular the facets of competence, dutifulness and achievement striving were all positive predictors of the motivation to learn"
 * 8) This section could possibly be abbreviated a bit, but it is key to the chapter. I would like to see some more introduction and conclusion emphasising what the main connections are between the Big 5 and Motivation.
 * 9) LOC
 * 10) Add http://onlinelibrary.wiley.com/doi/10.1002/job.416/pdf
 * 11) Eysenck
 * 12) Add File:Hans.Eysenck.jpg
 * 13) Other Personality Traits and Motivation
 * 14) I'd perhaps rename this as arousal and/or sensation seeking or something like that - to give it some more shape/focus.


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User:Brylie/Music Theory

The Just-In-Tone Music Theory class distils music theory to small bits and offers only the necessary elements for the learner to progress. It is important to keep the learner interested and confident in their ability to learn and apply their new skills.

The lessons are designed to give a progressive understanding of the elements of melody and harmony.

Seven Letters
The musical alphabet consists of seven letters:
 * A, B, C, D, E, F, G

Each letter is a label that we apply to musical notes. Although the letters are seemingly arbitrary they convey very specific melodic and acoustic information to anyone who learns to use them. Our goal is to learn to use these letters and other symbols and devices to communicate our ideas.

When you travel from any of the letters backwards or forwards you will eventually end up back where you started.

For example: C -> D -> E -> F -> G -> A -> B -> C

This full circle travel is called an Octave. The Cs at either end of the octave might have a different sound to them. One might sound higher or lower than the other but their qualities are the same.

Exercise
Spell any number of words using only these letters (A,B,C,D,E,F,G). The words do not have to be English; we are writing in the language of Music. Put your musical phrases in a safe place, e.g. a journal, as we will be using them again soon.

Chromatic
The seven letter musical alphabet is part of a 12 letter sequence called the Chromatic scale. The octave is a repeating pattern that we will be exploring in detail.

There are 5 more notes in the octave (12 notes) than there are letters (7 letters) in the musical alphabet. In order to denote some of the notes in the octave it is necessary to use extra symbols. There are two symbols commonly used called sharp (#) and flat (♭). Here are the 12 notes of the octave represented as letters: Notice that the only letters that do not have a sharp (#) are B and E. Thus, there is no note between B and C or E and F.

{{Hide in print|

Quiz
{ How many letters are there in the musical alphabet? { 7 _3 }
 * type="{}"}

{ How many notes are there in an Octave? { 12 _3 }
 * type="{}"}

{ The { Chromatic|chromatic _10 } scale contains all of the natural (A-G) and enharmonic (sharp/flat) notes.
 * type="{}"}

{ Generally, there are no sharp (#) nodes directly after { B | b _2} and { E|e _2}.
 * type="{}"}

}}

Intervals
Musical intervals describe how many chromatic notes it takes to go from one note to another. For example, to move from the note of C to the note of E you pass through 4 notes:

C -> C#, D, D#, E

The interval, of 4 notes, from C to E is called a Major third. Lets look at some additional intervals and their names:

 * minor 2nd (m2) = 1
 * Major 2nd (M2) = 2
 * minor 3rd (m3) = 3
 * Major 3rd (M3) = 4
 * Perfect (Major) 4th (P4) = 5
 * Flatted 5th (b5) = 6
 * Perfect (Major) 5th (P5) = 7

Triads
Triads are created by combining three notes together. There are common triads that the learner will encounter as a foundation to more advanced chord theory. The triads covered initially are Major (M), Minor (m), and Diminished (dim).

 * m = 1, b3, 5 (1, m3, M3)
 * M =  1, 3, 5 (1, M3, m3)
 * dim = 1, b3, b5 (1, m3, m3)

Diatonic Progressions
The Major, Minor, and Diminished triads comprise the chords found when you create progressions using only diatonic scale tones. The sequence of chords that emerges is as follows:
 * M  m    m   M  M m    dim

Additional Intervals

 * minor 6th (m6) = 8
 * Major 6th (M6) = 9
 * minor 7th (m7) = 10 (Dominant 7th)
 * Major 7th (M7) = 11
 * Octave = 12
 * Unison = 0

Seventh Chords

 * Minor 7th (m7) = 1, b3, 5, b7
 * Major 7th (Maj7) = 1, 3, 5, 7
 * minor 7th flat 5th (m7b5) = 1, b3, b5, b7

Triads are the base of these seventh chords:
 * m (1, b3, 5)
 * m7 (1, b3, 5, b7)

 * M (1, 3, 5,)
 * M7 (1, 3, 5, 7)

 * dim (1, b3, b5)
 * m7b5 (1, b3, b5, b7)

Diatonic Sevenths
When you construct a progression of seventh chords using only tones from the diatonic scale, the following pattern emerges:
 * M7 m7 m7 M7 7 m7 m7b5

Modes
The Ionian and Aeolean modes, while being the same (just rotated), serve as the lenses that we use to analyze tonal music. It is important to explore these modes early and often in the course of study.
 * Ionian (221222 / WWhWWW)
 * Aeolean (222122 / WWWhWW)

Function

 * Numeric analysis
 * Transposition

Modulation

 * Changing chord function
 * Secondary dominants
 * m7b5 → Dom7

title: User:Brylie/Music Theory


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User talk:Theresa Thomas

 Hello Theresa Thomas, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 07:17, 5 November 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 5 November 2010 (UTC)

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User talk:MNaeger

 Hello MNaeger, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 07:17, 5 November 2010 (UTC)

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text:
Motivation and emotion/Textbook/Emotion/Music/Styles

Jazz
Jazz is a musical genre which originated in the Southern American states in African-American communities at the beginning of the 1900’s. It draws influence from a number of European and African traditions with its early development incorporating 19th and 20th Century American popular music. Since then it has spawned a plethora of sub genres due to its somewhat free form and expressive nature, and ability to incorporate a number of cultural influences in to its varying styles. It is also noted as drawing influence from a number of blues and gospel influences, with some of the most recognisable artists and proponents of jazz music including Miles Davis, John Coltrane and Charles Mingus. From latin jazz, jazz fusion, jazz funk, big band style swing jazz, be bop, cool jazz, soul, traditional European jazz, and swing they all have a profound influence on human emotions and evoke a certain feeling in those who enjoy this style of music. More information: Beginnings of European jazz

Rock
Rock is probably the most popular form of “popular music”, and assumed to be the one with the largest following both in those who express it and those who choose to listen to it. It is estimated that it entered mainstream listening around the mid 1900s’ (roughly 1960s) and draws its base influence from 1940s and 50s rock and roll and blues. One of the primary reasons rock entered the mainstream roughly during the 1960s is because this was the dawn of the electric guitar, a period in time where its invention was relatively new and also a time when its popularity shot to new levels. As each decade passed, new genres of rock were invented, and bended with other forms of music such as jazz (creating jazz rock fusion), and further incorporated influences from Latin, soul and funk music. The period of time during the late 1960s and into the1970s and 1980s saw the usage of rock elements and the electric guitar soar to new levels, with new exciting styles and sounds of rock emerging. Some of them were heavy metal, hard and alternative rock, soft rock, progressive rock, and the timeless and iconic style known as punk rock. This era saw the emergence of such bands such as Metallica, The Ramones, Guns and Roses, Van Halen, The Rolling Stones, The Who, Fleetwood Mac, Led Zeppelin and Black Sabbath. However, one of the most famous figures to emerge from these eras was none other than Jimmy Hendrix, who typically expressed himself through various forms of Psychedelic rock and Blues. His unique style of playing, musicianship, and raw ability saw him become one of the greatest guitarists of all time, and cemented his place as one of the founding fathers of rock today.

Classical
Classical Music is one of the oldest forms of music, and has estimated to have been around since the 11th century however the term "classical music" has only been around since the 19th century which became known as the golden age of classical music and saw the likes of composers such as Johann Bach and Beethoven.

Classical music encompasses a varying array of forms, ranging from symphony, concerto, sonata, symphonic poem and even dance music. Classical music is recognised by its complex relationship between its emotional qualities and the very complex and intellectual means by which it is achieved. The varying forms of classical make use of the proces by which the idea or motif of the song is repeated in differing contexts, this is known as musical development. Composers of classical music typically have a strong desire to achieve a high level of technical achievement in the writing of their music, with the performers also expecting to have a very high degree of technical mastery. Classically trained musicians are expected to have a significant level of technical mastery in order to play the music, with skills such as sight reading, ensemble playing, tonal and harmonic property understanding, and familiarisation of the musical idiom from the given period of music they are performing. A thorough knowledge of performance practice is also an essential skill

Classical Music is divided into 5 era's, they are
 * Medieval Era
 * Renaissance Era
 * Baroque Era
 * Classical Era
 * Romantic Era


text: File:2010 11 05 07 46 02.djvu

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Numerical Analysis/Computing the order of numerical methods for ODE's

Introduction
The subject of this analysis is the order of accuracy of numerical methods for solving ordinary differential equations. When we mention "order of accuracy" for a particular numerical method, we usually mean the order of the global truncation error. The global error is the cumulative error in the numerical solution that is produced on an interval that we need the ODE to solve on. A local truncation error is the error in the numerical solution (the round off errors produced by computing are excluded) generated at a particular step, when the previous step solution is considered exact (although it is not exact, unless the previous step is the boundary or initial condition). The local truncation error is usually denoted as $$O(h^{(n+1)})\,$$, where n is the order of accuracy on the entire interval of the differential equation and h is the step size (for fixed step size discretization). The importance of the order of a numerical method is reflected in the fact that we need fewer steps to numerically solve an ODE with a prescribed error tolerance, if we use a higher order numerical method. This issue becomes very important when we try to solve an ODE that describes a relatively fast changing physical proccess, such as an internal combustion in the IC machines, vibrations in structures, etc.

The following two examples show how to determine the order of a numerical method. The first example is about a set of Runge Kutta methods of the second order. It shows how we can derive the conditions on the parameters that we introduce in the general form of the method and the possible choices for the parameters. The second example can be used as an exercise, since some steps are hidden and can be seen by selecting the marked button to show the missing steps that complete the derivation. The example shows how the conditions are derived for a Runge Kutta type general form with slope evaluations at three different locations within the current step. The main discussion is about the comparison of the Taylor series expansion and the corresponding numerical method recurrence equation. Based on this comparison, term by term, we conclude on the order of the local truncation error, which is then used to make a statement about the order of the method. The examples are followed by a concept quiz, which is convenient for a user to review the theoretical details that are used in the examples. Finally, we end up the analysis with a conclusion.

Example 1. Determination of the parameters to establish a second order Runge Kutta method
Let a single step numerical method for solving ODE of the type

be given by the following [2,4] recurrence formula

$$ y_{n+1}=y_n+h(a_1k_1+a_2k_2) \,$$

with

$$k_1=f(t_n,y_n)\,$$

$$k_2=f(t_n+ph,y_n+hqk_1)\,$$

Using Taylor series expansion about tn, the following can be obtained

Assuming that the previous point in (1.2) is exact (which we may because we are analyzing the local truncation error, i.e. the error generated at the last step), we will denote $$ y(t_n)=y_n, y(t_n+h)=\overline{y_{n+1}}\,$$. Therefore, both $$y_n\,$$ and $$\overline{y_{n+1}}\,$$ are considered exact values of $$y(t)\,$$ at $$t=t_n\,$$ and $$t=t_n+h\,$$, respectively.

Then,

Using the differential equation (1.1) and replacing the derivatives of y, (1.3) becomes

where

or in the compact form (1.5) is $$f'(t_n,y_n)=\left (f_t+f_yf) \right)_{t_n,y_n},\,$$

and

or in the compact form (1.6) is $$f''(t_n,y_n)=\left ( f_{tt}+2f_{ty}f+f_tf_y+f_{yy}f^2+f_yf_yf \right )_{t=t_n,y=y_n}\,$$

After substituting the expressions for f′ and f″ into the Taylor series expansion (1.4), the following is obtained

Now, our objective is to adjust the method's recurrence equation (1.2), such that it can be compared, term by term, with the equation obtained using the Taylor series expansion (1.7). This step requires the Taylor series expansion of the $$ f(t_n+ph,y_n+hqk_1)\,$$ term about the $$(t_n,y_n)\,$$ point, as follows.

or in the compact symbolic form (1.8) is:

By substituting (1.9) into (1.7), the following is obtained

Finally, the method's equation (1.10) and the exact one step solution (1.7) can be compared. By substracting (1.10) from (10.7), the following error expression is obtained.

where

It can be noticed that the multiplier expression of $$h^3\,$$ in (1.12) does not cancel out in (1.11) for any combination of the parameters, since there are terms that appear only once without any parameter. This means that the dominant term in the error expression cannot be of higher order than $$O(h^3)\,$$. In order to achieve the local truncation error of the third order, all terms in the error expression containing $$ h^0, h^1, h^2\,$$ must be eliminated, by adjusting the 4 parameters. By satisfying the equations (1.13-1.15), the local error is of $$O(h^3)\,$$, and, consequently, the global error is of $$O(h^2)\,$$ order. Therefore, the resulting method is a second order method.

The equation set that must satisfied is the following

There are three equations (1.13,1.14,1.15) but with four unknown parameters. This means that one parameter must be chosen. Now, it is tempting to ask whether the extra parameter (additional degree of freedom) could be used to cancel next term in the error expression. However, we have already seen that it is not possible in this case, due to the additional term that are not associated (multiplied) with a parameter.

Since one parameter can be chosen, then there is non-unique form of a second order method.

 * (case 1) Choose $$a_1=a_2\,$$, then $$a_1=a_2=1/2, p=q=1\,$$

The method equation (1.2) for these parameter becomes

$$ y_{n+1}=y_n+\frac{h}{2}(k_1+k_2) \,$$

with

$$ k_1=f(t_n,y_n)\,$$

$$k_2=f(t_n+h,y_n+hk_1)\,$$

or in a compact form

$$ y_{n+1}=y_n+\frac{h}{2}(f(t_n,y_n)+f(t_n+h,y_n+hf(t_n,y_n))) \,$$

 * (case 2) If $$ a_1=0\,$$ is chosen (notice that $$ a_2=0\,$$ cannot be selected, because of the last two equations), then

$$ y_{n+1}=y_n+hk_2 \,$$

$$k_1=f(t_n,y_n)\,$$

$$k_2=f(t_n+1/2h,y_n+1/2hk_1)\,$$

or in a compact form

$$ y_{n+1}=y_n+hf(t_n+1/2h,y_n+1/2hf(t_n,y_n)) \,$$

Example/Exercise 2. A single step ODE numerical method order computing with three slope evaluations (Runge Kutta 3-rd order)
Let the recurrence equation of a method be given by the following of Runge Kutta type with three slope evaluations at each step

with

$$ k_1=f(t_n,y_n),\,$$

$$ k_2=f(t_n+p_1h,y_n+q_{11}k_1),\,$$

$$ k_3=f(t_n+p_2h,y_n+q_{21}k_1+q_{22}k_2),\,$$

Taylor series expansion of $$y(t_n+h)\,$$ about $$t_n\,$$ is the same as in Example 1. Therefore, we will just use the final expression (1.7), since the procedure of the derivation is the same. For convenience, the final expression is repeated, which is going to be a reference equation for the comparison with the method's recurrence equation. Since the formulas for the given form of recurrence equation will get complicated, we will use the compact symbolic notation for the derivatives, which is shown in Example 1.

The Taylor expansion of the terms in (2.2) is shown up to $$O(h^4)\,$$, rather then up to $$O(h^5)\,$$, as we should in order to check that eventually next higher order terms cancel out, but we will assume that the method cannot achieve better local accuracy then fourth order, or equivalently, the global error of the third order. This will save us getting into the third level expansion of the two variable function f, which has 18 terms and would not be appropriate due to its length (even if the compact symbolic notation is used).

After we prepared the Taylor series expansion, we need to adjust the method's recurrence equation such that it can be compared with the Taylor series (2.2).

The Taylor expansion (equation 2.3)

$$y_{n+1}=y_n+ha_1f(t_n,y_n)+ha_2(f+p_1hf_t+q_{11}hf_yf+\frac {h^2}{2}(f_{tt}p_1^2+f_{yy}q_{11}^2f^2+2f_{ty}p_1q_{11}f)+O_2(h^3))+\,$$

$$ha_3(f+p_2hf_t+f_y\left(q_{21}hf+q_{22}h(f+p_1hf_t+q_{11}hf_yf+O_3(h^2))\right)+\frac {1}{2}(f_{tt}(p_2h)^2+f_{yy}h^2(q_{21}f+\,$$

$$q_{22}(f+\underbrace{p_1hf_t+q_{11}hf_yf+O_4(h^2)}_{O_5(h)}))^2+2f_{ty}p_2h^2(q_{21}f+q_{22}(f+\underbrace{p_1hf_t+q_{11}hf_yf+O_4(h^2)}_{O_5(h)}))))\,$$

Now, we need to group the terms in the similar way they are grouped in the Taylor series (2.2), such that we can establish the conditions on the parameters that will yield the same terms as in the Taylor expansion up to the terms containing $$h^4\,$$.

By comparing the two expressions (2,4) and (2,2), the following system of equations is obtained.

At the first glance, the system is closed, the number of equations is (2.5 through 2.12) which matches the number of undetermined parameters. However, only 6 equations are independent, the rest of them can be obtained from those 6 equations. By dividing (2.10) with (2.12), we can obtain that $$q_{11}=p_1\,$$. Similarly, by substracting (2.11) from (2.9) equation, we see that $$p_2=q_{21}+q_{22}\,$$. When we replace these two results into the rest of the equations, it is evident that the (2.6) and the (2.7) are the same, and (2.8) and the (2.9) equations are the same. Therefore, two equations can be obtained from other six, and we have to choose two variables in order to obtain a solution for the parameters.

For example, we can choose that $$p_2=1, q_{11}=1/2\,$$, then we obtain the following recurrence equation.

where

$$k_1=f(t_n,y_n)\,$$

$$k_2=f(t_n+1/2h,y_n+1/2hk_1)\,$$

$$k_3=f(t_n+h,y_n-hk_1+2hk_2)\,$$

The recurrence equation (2.13) is known Runge Kutta third order method [1,3] (List of Runge–Kutta methods), which indicates that our approach was correct.

Quiz - Method order computations, local and global truncation error
{What is the main importance of having a higher ODE method order -It always better to increase the accuracy of the solution +It requires fewer steps for the same accuracy restriction -We need the method to be better consistent -We need the method to be more stable -It provides better margin of stability
 * type=""}

{The number of slope predictions (number of k's) in an RK method is related to the order of the method as follows. The order of the method is:
 * type=""}

-exactly equal to the number slope predictions -equal to number of k's (slope evaluations) +1 +upper bounded by the number of k's -not related to the number of k's

{An n-th order of the method means that -the global truncation error is of the order $$O(h^{n+1})\,$$ -the local truncation error is of the order $$O(h^{n})\,$$ +the global truncation error is of the order $$O(h^{n})\,$$ -the maximum of the global and local error is n-th order
 * type=""}

{The order of the local truncation error is related to the global truncation error in the following way -they are about equal -they are not related -the global is one higher order than the local +the local is one higher order than the global
 * type=""}

{Is the following true? The global truncation error is always less then the local, since the errors partially cancel out over steps. -true +false
 * type=""}

{What is the highest order that we can achieve with a Runge Kutta type method by using 5 k's? -We can achieve 5-th order -That method is not used, since it is not consistent. +We can achieve 4-th order -We can achieve even higher order, it is sixth order accurate.
 * type=""}

{How is the order of single step methods related to consistency of the method? +At least first order of the method indicates that the method is consistent -To be consistent, a method must be second order or higher -not related at all, a single step method can be inconsistent and be of any order.
 * type=""}

{What is the main "tool" to prove the order of a numerical method for solving ODE's of type $$y'=f(t,y(t))\,$$? -Solving a specific "stiff" problem $$y'=f(t,y(t))=-Ky\,$$ and showing that it converges for specific real positive K -Taylor series expansion for one variable +Taylor series expansion for one and two variables -Solving the ODE equation analytically and then numerically
 * type=""}

{If we show that all terms in a numerical method for ODE recurrence equation match the terms in the Taylor series expansion up to the terms with $$h^n\,$$, but we do not analyze whether further cancellation of the terms exist, the following is true -The method is of the order n -The method is of the order n, since the global error is of the order $$O(h^{n})\,$$ +The order of the method could be higher than n-1, but it is at least n-1
 * type=""}

{If we use an explicit n-th (n>1) order accurate multistep method with s steps (s>2) to solve an ODE, but we use an n-1 order method to calculate the missing initial points that we need to start using the multistep method, what is the global order of accuracy of our calculation? -There is no such combined method, we cannot do that -The overall method is n-th order, since we just use the n-1 order method for few initial points +The overall method is of the order n-1, since the initial error from the lower accurate method remains in the cummulative error (global) -The order of the method could be either n or n-1 -The order of the method is something between n-1 and n, just like the RK 45 method
 * type=""}

{What do we get as an order of error from the following expression

$$\frac{1}{h}O(h^2)+10O(h^3)\,$$?

 * type=""}

-Something between $$O(h^2)\,$$ and $$O(h^3)\,$$ -$$O(h^2)\,$$ -$$O(h^3)\,$$ +$$O(h)\,$$

{What do we get as an order of error from the following expression

$$sin(h^2)\frac{1}{h}O(h^2)\,$$?

 * type=""}

-Something between $$O(h^2)\,$$ and $$O(h)\,$$ -$$O(h^2)\,$$ +$$O(h^3)\,$$ -$$O(h)\,$$

{What do we get as an order of error from the following expression

$$e^hcos(h)\frac{1}{h}O(h^2)\,$$?

 * type=""}

-It must be of order $$O(h^2)\,$$ -$$O(h^3)\,$$ +$$O(h)\,$$

Conclusion
The main reason that we need to consider the order of a numerical method for solving ODE's is that it directly determines the step size i.e. the number of steps that we need to calculate the recurrence equation for. This directly influences the amount of computational work to obtain the prescribed accuracy.

The main approach for computing the order of a numerical method is that we first use Taylor series expansion for the exact differential equation to obtain the next value, namely $$y_{n+1}\,$$ using the previous value $$y_n\,$$ and the right hand side function $$ f(t,y(t))\,$$ evaluated at $$t_n\,$$. We cannot say in advance up to which order we need to expand those terms in the Taylor series, since we are solving for the order of the method. The general rule is to expand the terms one order higher that we expect the method order is.

The second part of the method order computing is to write all terms in the given method recurrence equation in terms of the functions f and y evaluated at $$t_n\,$$. Hence, the function f appearing in the recurrence equation as an evaluation at time different than $$t_n\,$$, need to be replaced with a truncated Taylor series expansion about $$(t_n,y_n)\,$$.

The third part is to compare those two expressions obtained in the two forementioned ways. The order with respect to the powers of h, up to which the terms from the adjusted recurrence equation match the terms in the Taylor series expansion, represent the order of local truncation error. After we computed the order of the local truncation error, the global truncation error is one order less than the local error order.

Since the order of a numerical method for ODE is, by the definition, equal to the global error order, we only need to compute the local truncation error order and substract one from it.

There are some methods that do not have preciselly determined order. It is usually the case with the predictor corrector methods, like the Runge-Kutta method 45.


text:
Latonic/Verbs

The Present Participle
The Present Participle is formed like this: They adapt to the gender of whatever they represent. The may be used in any of the following ways:
 * Regular verbs on -are => -ans: carmínans
 * Regular verbs on -ere => -ens: prennens
 * Regular verbs on -ére => -ens: temens
 * Regular verbs on -íre => -íens: finíens
 * Regular verbs on -íre => -ens: sentens
 * esse: (iéns), éns, essens
 * posse: possiens, possens
 * velle, nolle: volens, nolens
 * ire: íens
 * fêre: fíens
 * male: prennens, prennenti
 * female: prennenti, prennente
 * neuter: prennente, prennenta
 * Substantive (used litterally, similar to English word on -ing and -er, or as a profession): i cercanti (the searchers), te vide l'ambulans? (Can you see the one that is walking?), il videns (the seeing, as opposed to the blind), il carmínans, la carmínanti (the singer)
 * Adjective (similar to English word on -ing, or as a profession): te vide la dama carmínanti? (Can you see the singing woman?), il homo lavorans va esse compensato (the (=any) man who works will be rewarded), il pontifex battens (the warrior-priest), il carmínans iscrivens (the singer-songer)
 * Gerundive (replacing a whole sentence): cercante mi capu, me revigni a domu (I came back home because I was looking for my hat), carmínante me lavoro (I sing while I work). Note that this construction uses the dative form: carmínante for male (and neuter), carmínanti for female, other way around in the plural.

General
The Present Participle of regular verbs is formed like this: There are many irregular past participles (see below). Irregularities in the Past Participle often correspond to irregularities in the Perfectu. The Past Participles adapt to the gender of whatever they represent. The may be used in any of the following ways: Additionally, the PP is used for the Passive Voice: m'es visto (I am seen), l'es factu (it is being done), le fûi postu a lí (it was put there) The Passive Voice is also used with esse or stare for general, impersonal things: l' es/sta lavoratu a cuí (There is work going on here), le n'es nò sapítu facere istu (It is impossible to do that)
 * Regular verbs on -are => -ato: carmínato
 * Regular verbs on -ere => -uto: / (irregular: prenso)
 * Regular verbs on -ére => -uto: temuto
 * Regular verbs on -íre => -íto: finíto
 * Regular verbs on -íre => -íto: sentíto
 * male: temuto, temuti
 * female: temuta, temute
 * neuter: temutu, temuta
 * Substantive (rarely translated literally): il cercato (the one they searched), i persi (the lost ones, the ones that have lost the way, the ones we lost), l'amato/-a (the lover), il/la morto/-a (the dead one, the deceased)
 * Female Form (describing the literal action): su venuta/arrivata (his arrival), la questa (the request)
 * Adjective (similar to English words on -ed: l'adefíçu renovatu (The renovated building), il homo visto (the man you/we saw)
 * Gerundive (replacing a whole sentence): Finíto mi lavoro, mi me reposâi (having finished my work, I rested), Essuto/Ito/Stato, -a a ví, me vidi illu que l'intensi (Now that I was there, I saw what he ment).

List of irregular verbs
esse - essuto - been posse - possuto - allowed velle - voluto - wanted/willing to nolle - (noluto) - not wanting to ire - ito(like Group 2) - went/about to fêre - (factu)- been there, existed habíre - habíto/habuto - got, had, recieved biére (bevére) - biuto (bevuto) - drunk (ac)cadére - (ac)caduto - happened cedere - cesso - came, arrived queríre - questo - chosen, wanted cognere - connuto (cognuto, cogníto) - known, met squire - squito - knowing, found out sapíre - sapíto/saputo (seputo) - understood, been able to dicere (dire) - dicto - said devére (diére) - devuto, debto - had to, required, needed facere (fare) - facto - done, made valére - valso/valuto - costed messere - messo (messuto) - done movere - mosso/moto - moved nacere - nato, nacto, nêco - born placere - plaçuto (placuto) - pleased, glad parére - parso - appeared, seemed ponére - posto - put prennere - preso - taken rennere - reso - given back, returned manere - masto/maso - remained remanere (rimare) - remasto - stayed respondere - resposto - anwsered rompere - rupto (rûpto) - broken scrivere - scripto - written sidére - séso/siso - sat tenére - tenuto - held, had veníre - venuto (vinuto) - became, came vidére - visto - seen vincere - vincto - won vivere - vito/victo/vivo - lived, survived aperíre - aperto - appeared avertíre - avertíto - warned capíre - capto - caught, seased, captured costruíre - costructo - built duíre - ducto - lead moríre - morto (murto) - died, dead offeríre - offerto - given servíre - serto/servíto - served dare - dato - given stare - stato - been

title: Latonic/Verbs


text:
User:Egm6321.f10/Lecture plan of other courses

EML 5526 Finite Element Method: Spring 2011,

EGM 6341 Numerical Methods 1: Spring 2011, Spring 2010

EGM 6321 Principles of Engineering Analysis 1: Fall 2011, Fall 2010, Fall 2009

EGM 6322 Principles of Engineering Analysis 2: Spring 2009

EGM 4200C Aerospace Structures, Fall 2008

EML 4500 Finite element analysis and design, Fall 2008


text:
User:Egm6321.f10/Report table spring semester template

 Please keep this table and the wiki code tidy. Use exactly the same format that I created. Remember to log in before you link to your team's  archived  report. Only archived reports are accepted; please do NOT link to current versions. Links to the archived reports should be made within one hour of the deadline. Egm4313.s12 12:55, 20 December 2011 (UTC)

Inspiring video on teamwork and learning: A hole in the wall: How children learn without a teacher, (better video resolution from TED)

Lecture plan, Good model reports

[The best reports and grading comments]

= Teams 1 to 10 =

= Teams 11 to 20 =

= Teams 21 to 30 =

= Comments =

(a) : The deadline for any report is in general on a Wed, unless extended by formal announcement (e.g., to accommodate the travel schedule of students, etc.), or upon a reasonable request.

(b) : void for now.

(c) : 21:00 UTC = 17:00 EDT = 5:00 pm EDT = 4:00 pm EST. So there are 4 hours difference between EDT and UTC, and 5 hours difference between EST and UTC. From the second Sunday of March until the beginning of November, we use EDT. From the first Sunday of November, the Eastern time changes from EDT back to the EST, and we use 22:00 UTC = 5:00 pm EST.

 Report tables of other courses


text:
User:Eml5526.s11/Lecture plan

 Under construction throughout the semester. Egm6321.f10 21:49, 31 October 2010 (UTC)

= All versions =

EGM 6341, Spring 2011, Lecture plan

= Recorded lectures, TA user page =

Recorded lectures in E-Learning at UF

= Lecture notes, report table =

Additional presentations (video, wiki, static html) made in class may not be recorded on these transparencies.

A mtg number n followed by a version letter in parentheses ([a-z]) indicates that this pdf file had been updated with a 2nd version "b", 3rd version "c", etc. The second letter "x" after a version letter designates an extra lecture; this letter became "m" after the extra lecture had become a make-up lecture. If you had downloaded this pdf file before, you want to clear the cache of your browser so to get the new version.

djvu: (install the viewers DjView4 or evince) [[media:fe1.s11.mtg1.djvu|Mtg 1 (b)]], [[media:fe1.s11.mtg2.djvu|Mtg 2 (c)]], [[media:fe1.s11.mtg3.djvu|Mtg 3 (b)]], [[media:fe1.s11.mtg4.djvu|Mtg 4 (g)]], [[media:fe1.s11.mtg5.djvu|Mtg 5 (c)]], [[media:fe1.s11.mtg6.djvu|Mtg 6 (d,m)]], [[media:fe1.s11.mtg7.djvu|Mtg 7 (b,m)]], [[media:fe1.s11.mtg8.djvu|Mtg 8 (b)]], [[media:fe1.s11.mtg9.djvu|Mtg 9 (e)]], [[media:fe1.s11.mtg10.djvu|Mtg 10 (e,m)]], [[media:fe1.s11.mtg11.djvu|Mtg 11 (c)]], [[media:fe1.s11.mtg12.djvu|Mtg 12 (d)]], [[media:fe1.s11.mtg13.djvu|Mtg 13 (c)]], [[media:fe1.s11.mtg14.djvu|Mtg 14 (b)]], [[media:fe1.s11.mtg15.djvu|Mtg 15 (b)]], [[media:fe1.s11.mtg16.djvu|Mtg 16 (f,m)]], [[media:fe1.s11.mtg17.djvu|Mtg 17 (b)]], [[media:fe1.s11.mtg18.djvu|Mtg 18 (d)]], [[media:fe1.s11.mtg19.djvu|Mtg 19 (d)]], [[media:fe1.s11.mtg20.djvu|Mtg 20 (c)]] [[media:fe1.s11.mtg20.old.djvu|{old}]], [[media:fe1.s11.mtg21.djvu|Mtg 21 (c)]], Mtgs 22+23: Exam 1 (thus the Mtg no. from 22 to 26 should be incremented by 2, i.e., [[media:fe1.s11.mtg22.djvu|Mtg 22]] should be numbered Mtg 24, ..., [[media:fe1.s11.mtg26.djvu|Mtg 26]] should be numbered Mtg 28) [[media:fe1.s11.mtg22.djvu|Mtg 22 (a,m)]], [[media:fe1.s11.mtg23.djvu|Mtg 23 (c,m)]], [[media:fe1.s11.mtg24.djvu|Mtg 24]], [[media:fe1.s11.mtg25.djvu|Mtg 25 (b)]], [[media:fe1.s11.mtg26.djvu|Mtg 26 (c)]], (Mtg numbers are readjusted from Mtg 29 on) [[media:fe1.s11.mtg29.djvu|Mtg 29 (b,m)]], [[media:fe1.s11.mtg30.djvu|Mtg 30 (b,m)]], [[media:fe1.s11.mtg31.djvu|Mtg 31]], [[media:fe1.s11.mtg32.djvu|Mtg 32 (b,x)]], [[media:fe1.s11.mtg33.djvu|Mtg 33 (b)]], [[media:fe1.s11.mtg34.djvu|Mtg 34 (e)]], [[media:fe1.s11.mtg35.djvu|Mtg 35]], [[media:fe1.s11.mtg36.djvu|Mtg 36 (a,x)]], [[media:fe1.s11.mtg37.djvu|Mtg 37]], [[media:fe1.s11.mtg38.djvu|Mtg 38 (b)]], [[media:fe1.s11.mtg39.djvu|Mtg 39 (c)]], [[media:fe1.s11.mtg40.djvu|Mtg 40 (c)]], [[media:fe1.s11.mtg41.djvu|Mtg 41 (c)]], [[media:fe1.s11.mtg42.djvu|Mtg 42 (b)]], Mtgs 43+44: Exam 2

Report table

Course billboard

Inspiring video on teamwork and learning: A hole in the wall: How children learn without a teacher

Mourning the Death of Handwriting, By Claire Suddath. Time Magazine, Monday, Aug. 03, 2009.

Op-Art: The Write Stuff, by Inga Dubay and Barbara Getty, NY Times, 8 Sep 2009.

= References =

Books
$$\displaystyle \clubsuit$$ Fish, J., Belytschko, T.,  A First Course in Finite Elements , Wiley, 2007. amazon Student companion site (Chap 12, matlab codes, errata)

$$\displaystyle \clubsuit$$ Hughes, T.J.R.,  The Finite Element Method: Linear Static and Dynamic Finite Element Analysis , Dover, 2000. amazon

$$\displaystyle \clubsuit$$ Cook, R.D., Malkus, D.S., Plesha, M.E., Witt, R.J.,  Concepts and applications of finite element analysis,  4th edition, Wiley, 2004. amazon

$$\displaystyle \clubsuit$$ Johnson, C.  Numerical Solution of Partial Differential Equations by the Finite Element Method,  Dover, 2009. amazon

Web references
$$\displaystyle \spadesuit$$ Algorithm (wikipedia)

$$\displaystyle \spadesuit$$ Related MIT OpenCourseWare courses

$$\displaystyle \spadesuit$$ Wolfram|Alpha is ... "learning resource available to your students at no cost that works as a computational knowledge engine. Wolfram|Alpha is not a search engine like Google or Yahoo!, because unlike a traditional search engine, Wolfram|Alpha has the capability to instantly compute the answer to previously unasked questions instead of scouring the web and returning links to pages that already exist. The results are displayed in an easy-to-read, understandable format that can be used as a primary source for educational and academic purposes."

$$\displaystyle \spadesuit$$ Stephen Wolfram: Computing the theory of everything, video on TED.com

$$\displaystyle \spadesuit$$ EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.
 * 2nd-order linear ODEs
 * 2nd-order non-linear ODEs
 * ODE education

$$\displaystyle \spadesuit$$ ODE (Wikipedia): Be careful; always verify the sources.

Papers
$\displaystyle \color{red}{\clubsuit}$ NEW: Vu-Quoc, L., Zhang, X., and Walton, O.R., A 3-D discrete element method for dry granular flows of ellipsoidal particles, invited paper to the special issue of the Computer Methods in Applied Mechanics and Engineering on Dynamics of Contact and Impact Problems, Vol.187, No.3-4, pp.483-528, 2000. (pdf)

$\displaystyle \color{red}{\clubsuit}$ NEW: Vu-Quoc, L., and Zhang, X., An elasto-plastic contact force-displacement model in the normal direction: Displacement-driven version, Proceedings of the Royal Society of London, Series A, Vol.455, No.1991, pp.4013-4044, 1999. (pdf)

$\displaystyle \color{red}{\clubsuit}$ NEW: Plantard, G. and Papini, M., "Mechanical and electrical behaviors of polymer particles. Experimental study of the contact area between two particles.  Experimental validation of a numerical model.", Granular Matter, Vol.7, No.1, pp.1-12, 2005. (pdf)

$\displaystyle \color{red}{\clubsuit}$ NEW: Kane, G., String theory and the real world, Physics Today, Nov 2010.

$$\displaystyle \clubsuit$$ P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004.

$$\displaystyle \clubsuit$$ Nagasao, T. et al., Dynamic effects of the Nuss procedure on the spine in asymmetric pectus excavatum, The Journal of Thoracic and Cardiovascular Surgery, Volume 140, Issue 6, December 2010, Pages 1294-1299. Nuss procedure (wikipedia)

Demo of FEA commercial software
Abaqus: Solutions, Student edition ($100)

Ansys: Demo room, Products and simulations.

Comsol: Product tutorials, CFD simulation

MSC Software: FEA Learning Center, Patran and MD Nastran; Free student edition, Free curriculum kit (...a set of example problems that serve as tutorials ... user-friendly workshops ... designed to guide you step-by-step through all the features of MD Patran and MD Nastran and also include several real world application problems...)

News
$\displaystyle \color{red}{\clubsuit}$ NEW: Is openness in the digital space killing creativity?, Digital Planet, BBC, 15 Mar 2011.

$\displaystyle \color{red}{\clubsuit}$ NEW: 60-minute program Endless Memory, 19 Dec 2010: Part 1 , Part 2: Forgetting promotes the need to abstract various concepts.

$$\displaystyle \color{red}{\spadesuit}$$  NEW:

Bing, Now With More Wolfram, NY Times, November 11, 2009. ... Microsoft today announced that it will begin integrating statistics and other data from computational engine WolframAlpha into its search engine Bing.

$$\displaystyle \color{red}{\spadesuit}$$  NEW:

Ask.com to Return to Old Service, NY Times, November 9, 2010. ... Other search engines have taken on Google with little success. Cuil, one of the upstarts, closed in September after two years. But new challengers include Blekko and WolframAlpha.

$$\displaystyle \color{red}{\spadesuit}$$  NEW:

Fun With WolframAlpha, NY Times, June 5, 2009.

= One-dimensional problems =

Approximation
Commands for WolframAlpha: (Lagrange interpolation basis functions)

plot {{(x-1)(x-2)(x-4)} / {(3-1)(3-2)(3-4)}}, for 1 <= x <= 4

plot {{(y-2)(y-3)(y-4)} / {(1-2)(1-3)(1-4)}}, for 1 <= y <= 4

plot [{{(x-1)(x-2)(x-4)} / {(3-1)(3-2)(3-4)}} * {{(y-2)(y-3)(y-4)} / {(1-2)(1-3)(1-4)}}], 1 <= x <= 4, 1 <= y  <= 4

Commands for gnuplot:

splot [x=1:4] [y=1:4] ((x-1)*(x-2)*(x-4) / ((3-1)*(3-2)*(3-4))) * ((y-2)*(y-3)*(y-4) / ((1-2)*(1-3)*(1-4)))

Advection–diffusion equation
= Multidimensional scalar field problems =

Advection-diffusion equation
= Vector-field problem: Linear elasticity =

Three-dimensional problems
= Beams =

Minimum potential energy
 Lecture plan of other courses </b>

title: User:Eml5526.s11/Lecture plan


text:
User talk:NawlinWiki

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title: User talk:NawlinWiki