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User:B9 hummingbird hovering/Beacon of Certainty by Mipham/Topic1

The wisdom of the two modes of naming and having two titles
Many Dharma textual works have many names. When they are rendered in other languages like what has happened in English these many names form a veil of obscuration. To rent and view through this obscuration a deep understanding of the names and their etymology, nomenclature and orthography is mandatory. I had a profound realization that there is a value in having many different names within the tradition for one text, the value is that we do not become set on one fixed orientation and way of identifying a text. The different names inform each other and teach different people in different ways.

Topic One is given two titles that seed its narrative:
 * the first is addressed by the 'Mendicant' to the 'Sage'  as rendered into English by Pettit (1999: p.195): "According to which of the two negations do you explain the view?"; and
 * the second is identified in the table of contents of the Vărăṇasi edition of the Beacon published by the Institute for Higher Tibetan Studies which glosses it thus as redered by Pettit (1999: p.141) "Question 1:The basis as the coalescence of appearance and emptiness." (dri lan dang po gzhi snang stong zung 'jug)

The two negations are 'absolute negation' (med dgag) of the Geldanpas (the school of Tsongkhapa) and the 'implicative negation' (ma yin dgag) of an undefined 'other' school. Two truths by any other name? Pettit, I search in vain to source definitive definitions of his terminology and his interpretation of Mipham. They are there but they are embedded. Definitions are slippery and never adequate but necessary. I hold that like children it is always sound to give definitions room to move and space to play. Definitions and children are both living documents or should that be rewritten, living testaments. They define and redefine as they systemically involve and evolve. The Absolute Negation is within the semantic field of the 'inconceivable' (acintya) which when rendered into the Tibetan was the name given to the first major esoteric Buddhadharma institution documented in the Himalaya, Samye. That was just an aside tendered with salience. Another venture at an incisive note: the conventional pramana is 'inference' (anumana); the absolute pramana is 'direct-awareness' (pratyaksha). This is encoded in lexical contraction a powerful quality in the exegesis of the Tibetan Buddhadharma tradition. Beauford make explicit.

Pramana
Dudjom ''et. al.'' (1991: p.117 Enumerations) advise of 'Three Kinds of Valid Cognition' (tshad ma gsum):
 * 'direct perception' (mngon sum tshad ma; pratyakṣapramăṇa)
 * 'implicit inference' (dngos stobs rje dpag gi tshad ma; anumănapramăṇa)
 * 'scriptural authority' (lung/shin tu lkog gyur gyi tshad ma; ăgamapramăṇa)

In Buddhadharma as a whole direct perception and inference are acceded whereas scriptural authority is provisional and may be disregarded if it posits a challenge to the other two.

Inference and the syllogism in Buddhadharma is somewhat esoteric and impenetrable for most people but it is important to understand the distinction Mipham is making in certain section of Topic 1.

Anumana and the Buddhist syllogism
Having a working knowledge of the Buddhist syllogism is required to understand the finesse of aspects of Mipham's argument in Topic 1 otherwise the aspirant is likely to become confounded by the import.

 * 'subject' (dharmin; chos can)
 * 'probandum' (sădhya; sgrub bya)
 * 'reason' (hetu; gtan tshig)

Two Negations
The 'two negations' mentioned in one of the two titles for Topic 1 are 'absolute negation' (med dgag; prasājyapratiṣedha) of the Geldanpas which is a different orthographic rendering in English for the school of Tsongkhapa which comes from the Ganden Monastery established by him and where his body was entombed for many centuries prior to its desecration by the Chinese; and the 'implicative negation' (ma yin dgag; paryudāsapratiṣedha) of an undefined 'other' school. We may for expedience implicate the Jonangpa as the undefined 'other' school that sports a model of the 'implicative negation'. The Jonangpa with their Shentong teaching arising from the praxis of Kalachakra are the premier example. Shentong is generally paired in contradistinction with Rangtong. Rangtong is essentially negation with negative implication (our absolute negation) whereas Shentong is negation with positive implication which can be, for example, the Self of the Nirvana Sutra or the Sugatagarbha doctrine just to identify two.

Specifically, in the Beacon Mipham pipoints Shentong. Both the absolute negation and the implicative negation are elaborations of primordial purity according to Mipham. Where elaborations are ornamentations of the mind or conceptualizations. Primordial purity is 'inconceivable' (acintya) to the mind[stream] but evident to the Mind[stream]; where lowercase is obscured and that which is capitalized otherwise.

I find a useful visual metaphor for an absolute negative to be a representation of a 'blank slate' (tabula rasa) with an overwritten superimposition of negating cross identifying that the blank slate, iconic of an absence with an infinite potential, is negated, and the superscribed cross being the absolute to the negation, reinforcing that the representation of that which cannot be represented is bunk; and for the implicative negative to be a silhouette. The silhouette establishes an analogue of the negative in the absence of visual depth and the general lack of colour and the analogue of the positive or implicative aspect by the implication of features in relief from the neutral background, often white. The Shadow, as well in its full Jungian context is a useful intertextuality for the silhouette and our implicative negation and poignant given how a dogmatic reification of the implicative negative is seen in the personification of the Jungian Shadow in an example of objective demonization of a given subjects fears, anxieties and insecurities. Or even an arguably favourable construction of their 'tutelary' (yidam) from their hopes, dreams and aspirations. An interesting and culturally and metaphysically apt motif is the emergence of the Tulpa genre in the literary and visual arts.

"'The division of negatives, or negations, into affirming and non-affirming, or implicative and non-implicative, is traced to Mimāṃsā injunctions to refrain from activities that either imply another activity in its place or not. For example, a mountainless plain is an affirming negative that explicitly suggests or indicates a positive phenomenon (a plain) in place of its object of negation (mountains). Another type of affirming negative is one that by context suggests a positive phenomenon in place of its object of negation; for instance, being told that Shākyamuni was either a brahmin or kṣatriya (member of the royal or warrior class) and was not a brahmin suggests, by context, that he was of the royal caste. A positive phenomenon — being a kṣatriya — is suggested in place of the object of negation — being a brahmin — in this context.'"

The absolute negation is treated by Mipham in depth in his commentary to the Madhyamakavatara, the title of this commentary has been set into English by Duckworth (2008: p.232) as: Immaculate Crystal Rosary. The Immaculate Crystal Rosary with its protracted refutation and qualification of the absolute negative of Tsongkhapa is subsumed in the English work of the Padmakara Translation Group (2002). Which I have not yet had the favourable fortune of studying.

I appreciate that the first topic of the Beacon is a critique of Tsongkhapa and conveys the Two Truths and their associated pramana: conventional pramana and ultimate pramana. Moreover, conventional and ultimate pramanas are complimentary but mutually exclusive. Which is conveyed by Pettit and would have been conveyed by the teachers and commentaries with which he consorted. It may also explicitly stated in Mipham's text but I haven't got to it yet. But we know that the two pramana, the conventional and the ultimate will interpenetrate and coalesce. There are to be four joys in that. It also needs to be stated here that pramana by its nature requires validity. What is unstated but very important to this issue of these two types of negations is the mindstream that is the conduit of their engagement. If a mindstream which is a continuum or series perceives something with the sense bases and their powers, whatever is perceived is likewise a continuum. The cookie-cutter of consciousness vivisects all perceptions into series or continuums. If this is not understood then relative truth cognized is not valid. This is the upaya of Pratityasamutpada brought to ground. Whatever is a continuum is by definition 'neither one nor many'. All series are dependent and produced and therefore, not Ultimate. The Ultimate to be Absolute must be unitary and all-encompassing and unvivisected, unborn by both dualistic mindstream and form perceived by that same dualistic mindstream. A Mindstream though may be nondual. This is the 'noble' (arya) Ultimate pramana that perceives the Dharmata of whatever is 'directly experienced' (pratyaksha) and with which there is a unity of the perceiver, perceived and act of perceiving as enshrined and encoded in the sacred semiology of the Gankyil.

Relative or Conventional Truth
Shantarakshita in his synthesis of Yogacara-Svatantrika-Madhyamaka, distinguished two ways of understanding the relative truth, one that perceives the relative or conventional as consciousness or mind-only, and one that perceives the relative as 'habitual tendencies' (vasana). These views are mutually informing and resolvable and are not to be viewed as mutually exclusive. This view of the relative truth is further elaborated variously within the Nyingma school and particularly so in that of the Nyingma Dzogchen. Relative Truth is generally glossed or qualified by synonyms of 'appearance', (relative) 'perception' and 'apprehension' for example. Relative Truth or Conventional Truth is apprehended by, or mediated through, the Eight Consciousnesses (as is Absolute Truth or Reality which may also just be directly perceived solely with the Eighth Consciousness, pure awareness). The Eighth Consciousness, the 'Base' (gzhi) Consciousness is the continuum that continues through our dream-of-living. We are 'awakened' (bodhi) in the dream when the vāsanā are exhausted. The vasana are the Base, this is contentious but only when understood that the mindstream is constituted by vasana and the Mindstream is constituted by adhisthana and in truth there is no division between vasana and the blessings or 'the-empowering-flow-from-the-sacred'. The English rendering of 'the-empowering-flow-from-the-sacred' follows Namkha'i (1999, 2001) who renders it from 'byin labs', the Tibetan for 'adhishthana' (Sanskrit). In practice the mindstream becomes the Mindstream as we gather the 'two accumulations' (tshags gnyis) of 'merit' (punja) and 'wisdom' (jnana).

Some Dzogchen traditions and texts hold that the 'Base' (gzhi) and the 'All-Ground' (kun gzhi) are distinct, in others they are identified. The view of distinction is entertained by those who have little understanding of the historical, doctrinal, lineal, textual and sadhana relationships of Yogacara and Dzogchen. Or, it is a provisional teaching not to confuse. But they are just different ways of perceiving the sacred continuum-of-the-Ground[-of-being]. Where and how the vasana are 'perfumated' by the accumulated skillful and unskillful actions of body, voice, mind is treated differently in the tradition. However it is envisioned, the Seventh Consciousness has a stake in the tainting or 'perfumation' of the vasana which are housed in the All-Ground.

As an aside, the 'Eight Consciousnesses' (aṣṭavijñānakāya; rnam shes tshogs brgyad) of Yogacara are a development of the Abhidharma. The Abhidharma in the Himalaya was transmitted through the lens of Vasubandhu's Abhidharmakośa and its auto-commentary Abhidharmakośa-Bhāṣya both of which were translated into the Tibetan in the 8th century by Kawa Paltsek and the Indian pandita Jinamitra.

One way of perceiving the Relative Truth following Shantarakshita is of the mindstream constituted by the vasana which is deceptively experienced as external to the subject but is a projected interiority, a projection of the outflows into the conventional experience of the World determining our body, voice and mind, qualities and activities and therefore our Worldview and experience be it either valid or invalid permutations of the Conventional and Absolute. The Eight Consciousnesses and their body, voice, mind, qualities and activities are the Mindstream. There are many different finesses on these associations within the tradition but this provides a good working entry.

Ultimate or Absolute Truth
Discussed elsewhere. Ultimate or Absolute Truth in the English translations of the Buddhadharma from the Tibetan is increasingly being rendered Reality. For a sound introduction to the two divisions of the Absolute Truth whether: refer Bhavaviveka (endearingingly known as Bhavya) in particular and Svatantrika Madhyamaka in general. The Padmakara Translation Group (2005: p.386) provide a salient thumbnail overview where 'consequences' and 'consequential arguments' is an English rendering of 'Prasaṅga/Prasaṅgika': Bhavya holds that the consequential arguments of Buddhapalita are not on the same footing as those of Nagarjuna. In both cases, the consequences imply negations that could theoretically be formulated as positive (syllogistic) arguments. The difference between them is that, given what is known to be Nagarjuna's intention (the negation of all four positions of the tetralemma), his negations are to be understood as nonimplicative. But such a concession is not to be granted to the commentator [Buddhapalita in particular and all commentators of nonimplicative negations in general], whose task is to render explicit to the fullest extent the obscurities of the commented text. If the commentator uses consequences (unaccompanied by any positive and clarificatory statement), the resulting negations cannot automatically be regarded as nonimplicative. On the contrary, they are implicative and therefore undesirable in the Madhyamaka context...It is worth noting that it is in Bhavya that the important distinction between implicative and nonimplicative negations first appears.
 * 1) a conceptual construction mediated by the mind[stream], or
 * 2) a direct experience of the enlightened-Mind[stream]

'Nonimplicative negation' (Sanskrit: niṣedha) may also be rendered as 'existential negation'. 'Implicative negative' (Sanskrit: paryudāsa) may also be rendered 'predicative negation'.

Though implicative negations are for the most part undesirable in Madhyamaka, if a practitioner has direct experience of the Absolute and they render it into writing, a person who reads their document is encountering an implicative negation as it is a positive orientation of Madhyamaka's desirable negation. If an implicative negation becomes normalized in a 'practice community' (sangha) then it becomes a reification and collapses the spacious openness of Shunyata into dogmatism. At best such a reification may be envisioned as 'skillful means' (upaya) and becomes a rallying point of faith and induces certainty.

Base
The key to understanding the Two Truths in the Dzogchen view of both the Nyingmapa and the Bonpo evocations is the Base. 'Base' is a common English rendering of the Tibetan 'gzhi' which is a seminal technical term in Dzogchen Traditions of both Nyingmapa and Bonpo. The Dzogchen charioteers of the Nyingma borrowed the term 'gzhi' from Tibetan translations of Yogacara works in the Sanskrit and other languages where it was employed by Vasubandhu in his codification of the continuum of the Eight Consciousnessness which were a development of the Abhidharma. Indeed, Tibetan Abhidharma is through the lense of Tibetan renderings of Vasubhandhu from the Sanskrit, Chinese and other languages. Whether or not it was already evident in Dzogchen Bonpo philosophy prior to the entrance of the Buddhadharma into the Himalaya is yet to be determined.

The example I have provided of the 'outflows' (anushaya) of the 'Base' as perfumated vasana, comes into its fruition when we understand the Base in its Sanskrit analogue of 'ashraya' which may be rendered into English as 'shelter'. We take 'shelter' in the Base. This is primary. Those who have followed my weblog may be aware that the Essence (ngo bo), Nature (rang zhin) and Function (thugs rje) of the Base are the Three Greatness of The Awakening of Mahayana Faith which Wonhyo treats famously in his commentary which I have as yet been unable to secure in English.

This following is conjecture albeit it informed, but there is a view held by some scholars that some Dzogchen terma were translations of Chinese works and weren't given credit. Unscrupulous yes, but cultural conventions were different. The Chinese works may also have been codifications of ancient heart-advice of the Bonpo lineages. Discourse of the many scripts of Zhangzhung into English is just opening. The jury is out: actually, it hasn't even convened. Look with an open-heart without sectarianism or penchant for the Buddhadharma or any other name of a tradition. See the Great Knowledge beyond names. A friend told me that the term for 'Dzogchen' in the Japanese may be re-rendered back into English as 'Great Knowledge'.

A tree (much more fun than a Pot or a Pole) to a human is apprehended by the Six Sense Bases and their Powers and intermediated by the Seventh Consciousness to the Eighth. A tree to an ant is intermediated through their sentient-being-specific modes of perception. Is the tree perceived by the human and the tree perceived b the ant the same tree? This is unable to be determined. It is one of those fruitless questions that if asked the Buddha would have directed the aspirant to meditate. This example of the conventionality of the tree is also another way of envisioning 'emptiness' (that is empty of or an absence of self-nature or inherent-existence; nuhsvabhavata) of 'phenomena' (dharmas). The Abhidharma and the logico-epistemic traditions would have that the continuum by its nature is a flux and every moment of perception is unique. The one Mindstream never sees the one tree. We may model this by a traditional example of perception which is a series a continuum somewhat akin to beads on a mala. The beads are moments of the continuum that are made sensible by the reconstruction and decoding in our mind which organizes and filters information in a meaningful way, albeit untrue to Reality. All Conventional Truth is a simulacrum irrespective of whether it is valid or not. The tradition also uses another example that is excellent. That of a mindstream that perceives in a state of health and the same continuum of the the mindstream in a state of disease with jaundice. A white khata is perceived by the jaundiced mindstream as yellow. Neither the white nor the yellow colour are Reality, both are conventions of our sentient-being-specific modes of perception and decoding and reconstruction of them in our minds. Actually, it is more appropriate to the tradition to model the mindstream as a flowing out into the world. I don't know how that can be stated more appropriately in English yet at this time. But it is the 'out-flows' (anushaya) of Early Buddhism which were acculturated and recast from the sibling tradition of the Jain Dharma or the Sikh Dharma it was one of the two or even both.

The cookie-cutter consciousness is my gloss of all the consciousness bar the Eighth. We always perceive Absolute Truth we just don't recognize it as such[ness]. The Eighth Consciousness is immutable and all-penetrating and ever-present. The Eighth Consciousness is the Sun that always shines and the Seventh Consciousness are the clouds of adventitious obscuration that both arise as a result of our unskillful activities of body, voice and mind as well as inconceivably since before beginningless time. The Seventh Consciousness is effectively obscuration so it is the quality, refinement or the purity of this consciousness that determines whether valid Conventional or Relative Truth is cognized through the Six Senses and/or whether valid Absolute Truth or Reality is cognized by 'Rang rig pa' (Svasamvedana) through the modality of Direct Awareness (Pratyaksha).

There are many ways even in the Nyingmapa and Bonpo lineages of grappling the relationship of the Two Truths. It is important not to be seduced by orthodox theory but to realise the meaning beyond the words through practice, directly. For the Nyingma, the Two Truths tend to always coalesce and interpenetrate and in that way are one even though they are distinct. What determines Conventional Truth or Reality is the quality of the Mindstream.

Dudjom ''et. al''. (1991: p.105 Ennumerations) relate 'Two Aspects of Relative Appearance' (kun rdzob gnyis):
 * the correct relative appearance (yang dag pa'i kun rdzob; tathyāsaṃvṛti); and
 * the incorrect relative appearance (log pa'i kun rdzob; mithyāsaṃvṛti)

Examples
Unfortunately, we don't have the fantastic sky-flowers and the ephemeral cloud-castles of the sylph-like Gandharvas nor the fluffy-bunnies-with-antlars known to some as Jackalopes (and to others as the more sultry, ever-rapacious horny-hares) but the ever-popular traditional drudge of...pillars and vases ...the debating crowd goes wild! Yes, if you were wondering ,that was base humour dressed with the jaded smock of facetiousness. Thankfully, 'pillars' and 'vases' are not 'poles' and 'pots', though 'columns' and 'urns' would have been fresh. The other famous pervasive example through the Dharmic Traditions is the snake rope trick which is an example of valid and invalid cognition of Conventional Truth. All these examples are important in the literature and will be discussed as if they are understood then the point is identified.

In regards to the pillars and vases it should be born-in-mind that these examples are in their Dharmata-plenitude that is unborn-by-mind, the Lingam and the Bumpa which are analogues of the Phurba and Singing Bowl; Vajra and the Ghanta, Method (Upaya) and Wisdom (Prajna). All of these symbolic attributes are traditional personifications of enlightened qualities of Mind (note usage of capital). It should also be emphasized that Mind in Buddhadharma as it is oft-used is a reification of an essence. This rarely goes questioned, qualified or apologized! We tend to challenge those on principal in the Buddhadharma. But so many Dharma-writers spout Mind like their lexicon is rusted and seized. As such they are born by Mind[stream] atemporal. There is a play on the term 'born' in its split denotation of 'genesis' and 'birthing' as related though contradistinct with 'ported' and 'upheld'. The usage employed the latter as the former would compromise its primordiality. This entire semantic field of 'born' is evident in the human pregnancy as evident in our glorious womenfolk. 'Wisdom' as an English gloss for 'prajna' is ill-defined and I render it 'sapience'. The specificity of 'sapience' holds more quality and more bite in its semantic value than the more general milk-tooth term 'wisdom'.

Much ado about 'existence'
In Buddhadharma discourse we have all this talk of 'existence' which is just misdirection. In my practical experience the general Western understanding of the term 'existence' and the 'existence' of the technical term of Buddhadharma cover very little of the same semantic field. This qualitative difference is rarely reinforced to the Western audience. What is the value of this lexical choice except to confuse and marginalize people entering into the Buddhadharma? It is somewhat like the blunt rendering of 'Shunyata' as 'Emptiness' that is the mainstay of the stock literature. Every definition should reinforce that a literal meaning of 'shunya' is 'zero' and 'shunyata' is 'zero-ness'. The semantic field of 'shunyata' in the indigenous Sanskrit and 'zero-ness' to a non-technical virgin entrant to the Dharma I proffer would if not as blunt as our stock 'Emptiness' define a natural field of semantic congruence with the field of the indigenous Sanskrit. (I didn't nail that but I am going to put it on the backburner and it will come.)

Hookem holds (1991: p.30) "...there is a subtle shift or play occurring in the Sutras and commentaries between treating emptiness simply as a truthful statement about apparent phenomena and treating it as the nature of Reality itself."

To define a general working inclusive definition of Shunyata: 'An uncollapsed Openness that is permeated by Plentitude' is apposite. That is pure conjecture but it would be worthwhile doing linguistic fieldwork solidifying my unfounded premise founded on sapience. Nothing 'exists' in the general Buddhadharma definition that is dependent or conditioned. Why our dear lamas ever talk of such technical language to Westerners that is non-intuitive and which is precious-rarely defined is a mystery to this black-bird. Importantly, the quality of the mindstream determines the quality of the pramana. The quality of the sadhaka determines the sadhana not the initiated yana. Though the blessings of the initiate do leaven the mindstream letting the luminous-adamant of the Mindstream shine through. Given that it is appropriate that we flow to the next heading.

The interpenetration of Shentong & Rangtong
Pettit's rendering of Mipham's first topic in the Beacon challenges the view of Shentong. In other documents Mipham speaks for as well as against 'Shentong' which demonstrates his versatility and non-attachment. I tender that modeled on the Two Truths, an orthodox Nyingma view would be to see the 'interpenetration' and 'coalescence' (zung 'jug) of Shentong and Rangtong and be attached and predisposed to neither unless praxis made it so. That said, we need to appreciate that Shentong is not a homogeneity. There are a number of different envisionings of Shentong. I tender that all deep contemplatives, meditators and denizens of trance-modality are arguably a kind of Shentongpa.

Pratyaksha
Dudjom ''et. al.'' (1991: p.115 Enumerations) advise of 'Three Kinds of Direct Perception' (mngon sum gsum): Note that Dudjom ibid. identifies that the three are augmented by a fourth identified within canonical sources which is intellectual direct perception (blo'i mngon sum; manaḥ pratyakṣa).
 * the direct sensory perception (dbang po'i mngon sum; indriyapratyakṣa);
 * the direct perception of intrinsic awareness (rang rig mngon sum; svasaṃvedanapratyakṣa); and
 * the direct perception (of emptiness) by the yogin (rnal 'byor mngon sum; yogipratyakṣa).

A Wikipedia article on this is sorely needed:


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Structural analysis

Topic:Structural_engineering

Degrees of Indeterminacy
When analyzing structural models, the first question to ask yourself is whether the system is determinant or indeterminate. If you are looking at a truss, the first step is to count the number of unknown forces. This is the number of members (a single axial force for each member) summed with the number of unknown reaction forces. Second, count the available equations of equilibrium. This number is found by taking the number of joints and multiplying by 2 (two equations of equilibrium per joint). Finally, subtract the number of equilibrium equations from the unknown forces to find the degrees of indeterminacy.

Degrees of freedom
Degrees of freedom(DoF) In two dimension space, single part of structure have 2 displacements and 1 rotation.

In three dimension space, single part of structure have 3 displacements and 3 rotations.

In general, a rigid body in d dimensions has d(d + 1)/2 degrees of freedom (d translations and d(d −1)/2 rotations), 2 directions combine to 1 rotation, therefore use Binomial coefficient to determine number of rotations.

Energy method
energy method These are methods based on linear elastic behavior and conservation of energy, i.e. the work done by external forces equals the energy stored in the structure under load. Energy U = Fx/2 = F2 /2k where F is the applied force, x is the distance moved in the direction of the force at its point of application and k is the elastic stiffness of the part, again in the direction of the force at its point of application.

Influence line
Influence line is defined as the function which shows the variation of a structural quantity (shear force, bending moment, deflection etc.) due to the change in the position of unit load.

Virtual work
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies,[1] but they have also been developed for the study of the mechanics of deformable bodies.

Castigliano's method
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies,[1] but they have also been developed for the study of the mechanics of deformable bodies.

Flexibility method
In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns

Stiffness method
Stiffness is defined as the resistance to deformation. stiffness is inversely proportion to flexibility

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

title: Structural analysis


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Introductory Java/Lesson1/HelloWorld

[[File:software for making wallpaer.jpg]link title

title: Introductory Java/Lesson1/HelloWorld


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User:Arithmetix

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Arithmetix
hey I'm just some 50-ish guy wandered in off the net... nobody very very special, not especially a genius or anything... but I am somewhat special and I am a little bit smart. I'm reasonably well-educated, and there's something more: I decided a long time ago that I'd rather try and learn to think for myself than watch television.

Formal education: I did a technician's certificate in telephony, and followed up by lots of night school and was going to do electrical engineering but the way things went that didn't happen.

Informal education: I did, for one reason or another, end up with a lot of free time over many years and I used this for learning and teaching, and even some research, in a kind of mathematics I invented for myself and in a narrow little area of wave superposition.

I'm really into religion. Lots of people think I am weird. My friends tend to be very scary... extreme examples of one kind or another.

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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 9 November 2009 (UTC)

title: User talk:Kmani


text:
User talk:Jefs99~enwikiversity

 Hello Jefs99, 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 08:18, 9 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 9 November 2009 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Jefs99. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Jefs99~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:34, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:12, 19 April 2015 (UTC)

title: User talk:Jefs99~enwikiversity


text:
User talk:Pchapman12345~enwikiversity

 Hello Pchapman12345, 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 08:18, 9 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 9 November 2009 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Pchapman12345. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Pchapman12345~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:40, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:Pchapman12345~enwikiversity


text:
User talk:InformationKid

 Hello InformationKid, 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 08:18, 9 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 9 November 2009 (UTC)

title: User talk:InformationKid


text:
User talk:Largo~enwikiversity

 Hello Largo, 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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 * Chat with other Wikiversitans on #wikiversity-en.

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 08:18, 9 November 2009 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 9 November 2009 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Largo. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Largo~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:36, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:14, 19 April 2015 (UTC)

title: User talk:Largo~enwikiversity


text:
User:Ktindall

Questions from Debra R-W, based on the presumption that David and Robyn are staff developers or technicians.

 * Where do you start?

 * What are the drivers in your Institute for using online tools?

 * How do you evaluate what works well?

 * How do you test the usability of a tool prior to using it?

 * Do you have preferred tools that you encourage staff to use?

 * Do you up-skill staff to use the tools or do you set up the teaching materials on their behalf?

title: User:Ktindall


text:
Wiki Resources/Storyboard

Home | Learning Theories | Instructional Design & eAuthoring | ePortfolio | Storyboard | Wiki Analysis | Communication Tools | Resources Analysis | Group Work | Teaching Philosophy | Accessibility | Glossary

Introduction
The first procedure, before you start your learning object is to elaborate a storyboard. Through this useful tool you are organizing your ideas in relation to the project and at the same time filling the first ADDIE Model's phase called Analyse.

Learning Outcomes
At the end of this chapter the learner will be able to:
 * 1) Understand the relevance to build a storyboard.
 * 2) Understand how to construct a storyboard.
 * 3) Understand which the main characteristics in a storyboard.

We would appreciate if you give us your contribution in this project. If you have any query, suggestion or feedback, don't hesitate to contact us sending a message to our email: wikiresourcesgroup@gmail.com

Instructional Design Storyboards
Storyboarding is a method which was developed for use in film and animation and it is now also used for designing educational packages. In film, storyboards are developed before the movie starts filming and are used to help plan how each scene will look and what the key elements of the scene will be. The principal behind storyboards is the same for developing learning packages, except in most cases the storyboard looks less like a comic strip and contains more text. The storyboard defines structure of the package, page by page (or in the case of face- to-face teaching each panel could represent an activity). For each page, you need a fairly detailed description of what will go on that page. Practice varies from place to place, but some people will put the final text of the page into their storyboard while others will write a summary of what will go on the page; the choice is yours. The storyboard should include details of any images, interactions or other media that will be used in each page. This will either refer to an existing resource or describe what the resource will be. If someone else is going to be developing the resources, make sure you give them a good enough description to allow them to do this. Try not to have too many pages of just text; this would be quite boring for your learners. When developing your storyboard think about:
 * What do you want your learners to learn?
 * What skills do you want them to develop?
 * How can this package meet these needs?
 * What kind of activities would help?

The creation of a new learning package in a more formal manner allow educators more reflection about their current methods of teaching. In practice, storyboards are dynamic documents, and you will probably need to review them during the actual development of your package. This is quite normal; in fact, it is very rare for storyboards to remain static throughout the development of learning packages.

Watch here a short explanation by film-makers about storyboards.

Create a Content Outline
The high level objectives are the backbone of a storyboard creation. The high level objectives allow you to decide the number of lessons, topics and sub–topics the course can be broken down into. These objectives also allow you to decide what the approach within the topics would be. After defining the high level learning objectives, it is important to break the objectives into definite content outlines. Once the high-level learning objectives are in place, the outline of the course can be defined.

Read here an interesting article about the use of storyboard in education environment.

Visualize the course
Once the content outline is in place, it is time to visualize how the course should be. Content plays an important role in a course and it needs to be supported appropriately by use of relevant graphics. Graphics consist of static images, illustrations or animations that support the content. Never use a graphic element just to use up space. The graphic should be a visual representative of the content that you are explaining. While visualizing the placement of graphics in a course, it is by default that the graphics are placed on the left side of the screen and the corresponding content is displayed on the right side of the screen. This is for the easy readability factor.
 * Tip: For complex courses, it always advisable for the Instructional Designer to visualize the screens with the graphic designer, in order to achieve the desired result.

Use Interactive Elements
You are bound to lose the interest of a learner, if your course is devoid of any interactivity. Active involvement of the learner by the use of interactive elements and in-line assessments will hold the learners interest in the course. Roll-overs, click-and-reveal, multiple tabs can be different ways of representing content and graphics in the same space. Create scenarios if you would want to explain a situation to the learner. Similarly, if you are demonstrating a product that requires multiple steps to operate, create a simulation that will allow the learner to explore how that product works. These interactivities can be used depending on the complexity of a course that is being created.

Audio usage
When you use audio for a course, keep in mind that not all users may have an access to headphones or sound cards in their machine. Therefore, it is important that you do not cover any aspect in the audio that is not present on the screen. Keep in mind to retain important instructions and directions on-screen even if they are being narrated. For example: Click on the right posture a presenter needs to have while making a presentation. Avoid using a narration in in-line texts or assessments. They do not add any value in such screens. When describing simulations, it is always better to list the steps also apart from the narrations being used. In case there more than 15-20 steps involved to complete a simulation, it is better to have the steps listed down for an easy reference, rather than forcing the learner to click on the replay button.

Use assessments
Use assessments at regular intervals to test the learners instead of having a cluster of questions towards the end of the course. This helps to reinforce the information/ knowledge you are trying to impart to the learner, how well it is being processed & understood by the learner and most importantly it is also an indicator of how successful the course is. Ensure that the assessments are in tandem with your learning objectives you have defined at the start of the course.

Summarizing
Listed below are some additional tips to keep in mind while creating a storyboard in e-learning: (Pratibha G, 2009)
 * 1) Consider and advocate for the learner's point of view
 * 2) Be a learning expert, not a subject matter expert
 * 3) Create learning objectives that will allow the learner to take away an understanding from the course
 * 4) Ensure that the content maps to the learning objectives
 * 5) Ensure that the flow of learning objectives and content is logical
 * 6) Apply an instructional strategy
 * 7) Design the course appropriately
 * 8) Do not clutter the screen with an excessive use of graphics and content. The neater the screen, the better is design of the course and easier for the learner to understand the course.

See here a presentation about storyboards.

The WikiResources Group Storyboard
Our group decided, after many discussions and analysis about pros and cons, to present two kinds of storyboards. Firstly a common form built in Microsoft Power Point containing a series of steps until the last design about the project. Click here to access the Storyboard in Power Point presentation. An alternative way to create a storyboard was tried, called storybird, a experiment which we can create the storyboard as a story.

You can visualize the result of our experiment here.

title: Wiki Resources/Storyboard


text:
User:Sensei 0208

 * Hello and welcome!

title: User:Sensei 0208


text:
User:B9 hummingbird hovering/Beacon of Certainty by Mipham/Topic1/sky-flowers

User:B9_hummingbird_hovering/sky-flower

Sky-flowers
kūge 空花 ke 空華 kokūge 空華 (the first character is noise) :-(

The motif of sky-flowers is a device used in Buddhadharma poetry, logic and iconography.

In Buddhadharma the sky-flower is a metaphor and a teaching tool for something, a 'thing', that is beyond or independent of 'causes' (Sanskrit: hetu) and 'conditions' (Sanskrit: pratyaya). Therefore, a non-thing. If there was such a thing evident it would be an entity in the Buddhadharma, it would be an existent. An existent is something that is neither founded nor predicated nor dependent upon anything else in any way. An existent also takes into consideration the "Neither one nor many" logistic architecture. Generally in the Buddhadharma an 'existent' must be indivisible and not constituted by parts or segmentation. Segmentation either spatially or temporally compromises an 'existent'. Hence, if something that appears is conceived with the moving mind, the stream of the mind, the mindstream, then it is not 'existent'. When the mind is still and perceives Dharmata or the Absolute through any of the senses, that is with enlightened mind[stream], this is 'unique' as it encompasses all causes and conditions and is beyond them. Dharmata and the Absolute is a unity, indivisible: where subject, object and the linking act of perception are a single, unreplicable experience, timeless.

Though a flower could be in the sky due to a number of causes the conditions are rare and unlikely that would keep one in that state. In truth, though possible as we can conceive of a possible world where there are sky-flowers it is unlikely in our current worldview and this is the import of the teaching. A possible world with sky flowers could be a world where an epiphyte is born in the air due to magnetic currents, somewhat like the possible physics of the Meissner Effect in support of the possible world of the Hallelujah Mountains in the fictional universe of the movie Avatar (2009). As an aside, the 'possible world' flora depicted in this movie is breathtaking. For an air-plant to be suspended in their air like the Halleluja Mountains would require though that the plant have some form of magnetic property. I do not know if there is any living plant with magnetic properties that could be suspended like a magnet in the Meissner Effect. This is just one an example. In another, a plant could be very light and aerodynamic and keep itself born at certain times of the day for example by the adventitious use of hot-air currents somewhat comparable to the employ of hot air currents by vultures to get their vast ungainly weight efficiently airborn. All that said, a sky-flower is whether possible or not, an anomaly. Flowers on Earth generally spring from plants which are terrestrial. I wonder if air-plants (Epiphytes) have flowers? They often look like a flower even though they are foliage. Sky flowers would be and are to be considered 'precious rare', rare as hen's teeth. That is blooming unlikely!

Followers of illusion
Khapushpa, flowers in the sky, spots before the eyes, Musca Volitantes; illusion, The Indian Hinayanists style Mahayanists, Shunyapushpa, S'UnyapuS.pa, sky-flower heretics, or followers of illusion.

...am i reading correctly that this was a pejorative term of the Hinayana for the Mahayana?

Jaina
The Jaina ... Yes indeed! A thing which is strictly in itself, which has absolutely nothing in common with all other things in the whole world, is a non-entity, a flower in the sky! If you wish to distinguish it from a non-entity you must admit 'Thingness' as a real Category, just as Causality and Substantiality.[32]

32. TS, kar. 1713 - tasmat kha-puspa-tulyatvam icchatas tasya vastunah, vastutvam nama, samanyam estavyam, tat-samanata.

A solid source for this quote

Bugger it doesn't have the abbreviations for the Jaina source in the Google Books. I may have this text as a PDF. I have one of Tscherbatsky's Logic textbooks. There might be a source in Internet Archives or Scribd. I know this text is in a million copies at Baillieu.

Poetry
"'At that moment of the fourth watch when the dawn came up and all that moves was not stilled, the great seer reached the stage that knows no alteration, the sovereign leader, the state of omniscience. When as the Buddha he knew this truth the earth swayed like a person drunk with wine. The four quarters shone bright with crowds of siddhas and mighty drums resounded in the sky. Pleasant breezes blew softly and heaven rained moisture from a cloudless sky and from the trees there dropped flowers and fruit out of due season as if to do him honor. At that time, just as in paradise, the mandarala flowers, lotuses and water lilies of gold and beryl fell from the sky and bestrewed the place of the Shakya sage.'" -  Ashvaghosa, Sanskrit poet, Quoted by Zoketsu Norman Fischer in his Talk Six

Ashvaghosa, now we are getting to her root of this... *hehehehe* a bit of a F-slip

Dogen with his bounty of natural metaphors employed the motif of sky-flowers often. Shobogenzo: ... we are all sky-flowers as are all Buddhadharma doctrines, we are but imaginations of ourselves... all teachings are upaya, no teaching is definitive, not even definitive sutra! Ephermeral and Adamantine entwined. Openness of Shunyata. Flowers are mandalas.

Bhavaviveka
Bhavaviveka

MadhyamakahRIdaya and TarkajvAlA

Dharmakirti
Dharmakirti (ca. 7th century)

Tillemans (1999: p.13) holds that: "We see quite clearly in the Buddhist logicians' use of inferences like the so-called bAdhakapramANa (based on DharmakIrti's VAdanyAya) that the example can be a nonexistent thing, like a rabbit's horn or a flower in the sky, and that the scope of the vyApti must therefore range over nonexistents as well as existents. Nor is it particularly infrerquent or revolutionary for Buddhists to give such nonexistent items as examples--BhAvaviveka, who was well before DharmakIrti, used the example of the sky-flower in MadhyamakahRIdaya and TarkajvAlA too and even DignAga used the example 'space' (AkAsha) in his Hetucakra, an example which is not actually existent for a Buddhist."

There is a lot more in the Tillemans book on Rabbit's Horns and SkyFlower.

Mipham
Mipham's 'The Sword of Wisdom' as rendered by Adam Pearcey (2004) of Lotsawa House: These appearances in all their rich variety Arise through dependent origination. Something that is truly independent, Like a lotus [in the sky], will not appear. (5)

ཇི་ལྟར་སྣང་བ་འདི་དག་ནི། ji ltar snang ba 'di dag ni/ རྟེན་ཅིང་འབྲེལ་འབྱུང་ཚུལ་གྱིས་སྐྱེ། rten cing 'brel 'byung tshul byis skye/ ལྟོས་པ་མེད་གང་ནམ་མཁའ་ཡི། ltos pa med gang nam mkha' yi/ པདྨོ་བཞིན་དུ་སྣང་མི་འགྱུར། padmo bzhin du snang mi 'gyur/

I placed the square brackets enclosing "in the sky" as though this may be understood and implied by the "lotus" and indeed nonsensical if the logical adage isn't known. That it is implied by Mipham demonstrates that he considered any reader of his text would be well versed in the literary pervasion of this pedagogic stock example. In the Buddhadharma and particularly Dzogchen philosophy and poetry, "sky" and "space" is attributed with the qualities of "immutable, unchanging" (Wylie: mi 'gyur). As the medium and placeholder, "sky", holds the space for the play of "dependent origination" (Sanskrit: Pratityasamutpada) which is the dance of Void (Sanskrit: Shunyata). This Mystery of seemingly endless changing of forms is the poetic and philosophical dance of creation and destruction of Naturaja, the dance of energetic movement and exchange, the immutable law of reciprocity.

Hagiography
Some legends say that when Gautama was born the earth shook, rivers stopped flowing, flowers fell from the sky, and a lotus flower sprang from the place where he first touched the earth....Beauford...source a source with more authority...

Iconography
Dunhuang and the Mogao Caves: "'The Mogao cave temples near the town of Dunhuang, at the edge of the Gobi desert in north-west China, are filled with one of the most extensive and exquisite collections of Buddhist paintings and sculptures in the world. Every surface of the walls and ceilings is covered with painted clay stucco, some 45,000 square metres in all: graceful acrobats of the sky scatter flowers and garlands, while dancers and musicians celebrate the beauties of the Buddhist Pure Lands; row upon row of miniature images of the Buddha, subtly varied in colouring or dress, adorn virtually every cave, and give the site its popular name of the Caves of the Thousand Buddhas. The Dunhuang caves remain one of the most perfectly preserved of the world's great religious sanctuaries.'"

http://books.google.com.au/books?id=DbxE8zOuRbUC&pg=PA5&lpg=PA5&dq=sky+flowers+buddhism&source=bl&ots=ozK1GN8jtz&sig=a1D30JghOTyLF7uJV8X7wTAqOkw&hl=en&ei=SQ_4SqiYA4WPkQXtnJiyAw&sa=X&oi=book_result&ct=result&resnum=7&ved=0CB0Q6AEwBg#v=onepage&q=&f=false


text: File:Turbulent.flow.characteristics.png

title: File:Turbulent.flow.characteristics.png


text:
User:Egm6321.f09.Team2/HW6

Homework Assignment #6 - due Wednesday, 11/18, 22:00 UTC

Problem 1
p.31-1 For Circular Cylinder Coordinates find:

1) $$(dx_i)\ in\ terms\ of\ (\xi x_j)$$

2) Find $$ds^2=\sum_i^{}(dx_i)^2=\sum_k^{}(h_k)^2(d\xi_k)^2$$ Identify $$(h_i)\ in\ terms\ of\ (\xi_j)$$

3) Find the Laplacian in curvilinear coordinates.

4) See Problem 8

Problem 2
p.31-1 For Spherical Coordinates find:

1) $$(dx_i)\ in\ terms\ of\ (\xi x_j)$$

2) Find $$ds^2=\sum_i^{}(dx_i)^2=\sum_k^{}(h_k)^2(d\xi_k)^2$$ Identify $$(h_i)\ in\ terms\ of\ (\xi_j)$$

3) Find the Laplacian in spherical coordinates.

Use Astronomy Convention

Problem 3
p. 31-2 Find the Laplacian (from problem 2) in spherical coordinates but use the Math/Physics Convention.

Problem 4
p.31-2 Show that the solutions for $$\lambda (\lambda +1)=n(n+1)$$ are $$\lambda = n\ $$ and $$ \lambda=-(n+1)$$

$$\lambda (\lambda +1)=n(n+1)$$

$$\frac{(\lambda +1)}{n} = \frac{(n+1)}{\lambda}$$

$$\frac{\lambda}{n}+\frac{1}{n} = \frac{n}{\lambda}+\frac{1}{\lambda}$$

Assuming n and $$\lambda$$ are sufficiently large, $$\frac{\lambda}{n} = \frac{n}{\lambda}$$

Cross multiply and $$\lambda^2 = n^2$$   or

 $$\lambda = n\ $$

Likewise, $$\lambda (\lambda +1)=n(n+1)$$

$$(\lambda)^2 + \lambda=(n^2)+n$$

$$(\lambda^2 - (n^2) = n - \lambda$$

$$(\lambda + n)(\lambda - n) = (-1)(\lambda - n)$$

$$\lambda + n = -1 $$

$$\lambda = -1 - n = (-1)(n + 1)$$

 $$ \lambda=-(n+1)$$

Problem 5
p. 31-3 Show that $$P_n(x)=\sum_{i=0}^{[n/2]}(-1)^i{(2n-2i)!x^{n-2i}\over 2^ni!(n-i)!(n-2i)!}$$ can be reduced to $$\sum_{i=0}^{[n/2]}{1\times 3\times...(2n-2i-1)\over 2^ii!(n-2i)!}(-1)^ix^{n-2i}$$

Problem 6
p.31-3 Verify that equations (1)-(5) on p.31-3 can be written as Equation (6)or(7) from p.31-3

As seen from Problem 5, Equation 6 and Equation 7 are equivalent; therefore, anything that can be written as Equation 6 can be written as Equation 7.

Eq. (1): $$P_0(x) = 1$$ Eq. (2): $$P_1(x) = x$$ Eq. (3): $$P_2(x) = \frac{1}{2}(3x^2 - 1)$$ Eq. (4): $$P_3(x) = \frac{1}{2}(5x^3 - 3x)$$ Eq. (5): $$P_4(x) = \frac{35}{8}x^4 - \frac{15}{4}x^2 + {3}{8}$$

can be written as: Eq. (6): $$P_n(x) = \sum_{i=0}^{\frac{n}{2}} (-1)^i(\frac{(2n-2i)! x^{n-2i}}{(2^ni!)(n-i)!(n-2i)!})$$

For Eq. (1), $$n=0$$ and $$i=o$$, so the summation is only 1 number:
 * $$ = (-1)^0(\frac{(0)! x^{0}}{(2^00!)(0)!(0)!}) = 1(\frac{1*1}{(1*1)(1)(1)}) = 1$$

For Eq. (2), $$n=1$$ and i goes from 0 to $$\frac{1}{2}$$.
 * $$P_2(x) = \sum_{i=0}^{\frac{1}{2}} (-1)^i(\frac{(2-2i)! x^{1-2i}}{(2^1i!)(1-i)!(1-2i)!} = x + \frac{i}{2*pi} = x$$

For Eq. (3),$$P_2(x) = \sum_{i=0}^{1} (-1)^i(\frac{(4-2i)! x^{2-2i}}{(2^2i!)(2-i)!(2-2i)!}) = (\frac{1}{2}3x^2) - (\frac{1}{2}) = \frac{1}{2}(3x^2 - 1)$$

For Eq. (4),$$P_3(x) = \sum_{i=0}^{\frac{3}{2}} (-1)^i(\frac{(6-2i)! x^{3-2i}}{(2^3i!)(3-i)!(3-2i)!}) = (\frac{1}{2}(5x^3) - (\frac{1}{2}(3x) = \frac{1}{2}(5x^3 - 3x)$$

And Eq. (5), $$P_4(x) = \sum_{i=0}^{2} (-1)^i(\frac{(8-2i)! x^{4-2i}}{(2^4i!)(4-i)!(4-2i)!}) = \frac{35}{8}x^4 - \frac{15}{4}x^2 + {3}{8}$$

Problem 7
p. 32-1 Verify that equations (1)-(5) on p.31-3 are solutions of the Legendre equation (1) on p.14-2 and p.30-4

Legendre equation is $$(1-x^2)y'' - 2xy' + n(n+1)y = 0$$

Eq. (1): $$P_0(x) = 1$$ Eq. (2): $$P_1(x) = x$$ Eq. (3): $$P_2(x) = \frac{1}{2}(3x^2 - 1)$$ Eq. (4): $$P_3(x) = \frac{1}{2}(5x^3 - 3x)$$ Eq. (5): $$P_4(x) = \frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}$$

So $$P_n(x) = y$$ and n is given in each equation,

Eq. (1): $$y = 1$$, $$ y' = 0$$, and $$ y'' = 0$$ with $$n = 0$$
 * $$ (1-x^2)(0) - 2x(0) + 0(0+1)1 = 0$$

Eq. (2): $$y = x$$, $$y' = 1$$, $$y'' = 0$$, $$n = 1$$
 * $$ (1-x^2)(0) - 2x(1) + 1(1+1)x = 0$$

Eq. (3): $$y = \frac{1}{2}(3x^2 - 1)$$, $$ y' = 3x$$, and $$ y'' = 3$$ with $$n = 2$$
 * $$ (1-x^2)(3) - 2x(3x) + 2(2+1)\frac{1}{2}(3x^2 - 1)$$
 * $$ 3 - 3x^2 - 6x^2 + 9x^2 - 3 = 0$$

Eq. (4): $$y = \frac{1}{2}(5x^3 - 3x)$$, $$ y' = \frac{15}{2}x^2-\frac{3}{2}$$, and $$ y'' = 15x$$ with $$n = 3$$
 * $$ (1-x^2)(15x) - 2x(\frac{15}{2}x^2-\frac{3}{2}) + 3(3+1)\frac{1}{2}(5x^3 - 3x)$$
 * $$ 15x - 15x^3 - 15x^3 + 3x + 30x^3 - 18x = 18x - 30x^3 + 30x^3 - 18x = 0$$

And Eq. (5): $$y = \frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8}$$, $$ y' = \frac{35}{2}x^3-\frac{15}{2}x$$, and $$ y'' = \frac{105}{2}x^2 - \frac{15}{2}$$ with $$n = 4$$
 * $$ (1-x^2)(\frac{105}{2}x^2-\frac{15}{2}) - 2x(\frac{35}{2}x^3-\frac{15}{2}x) + 4(4+1)\frac{35}{8}x^4 - \frac{15}{4}x^2 + \frac{3}{8} = 0$$

Problem 8
p.32-1 Continuation of Problem 1. Obtain the separate equations for the Laplace equation on circular cylinder coordinates an identify the Bessel differential equation. $$x^2y''+xy'-(x^2-\nu^2)y=0$$ Compare to King p.84

Given:
 * $$\nabla^2 \psi = \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial \psi}{\partial r})+\frac{1}{r^2}\frac{\partial^2 \psi}{\partial \theta^2}+\frac{\partial^2 \psi}{\partial \xi^2}$$

Assumptions:
 * 1) $$\psi \!$$ is not dependent on $$\xi \!$$
 * 2) The equation is separable, $$\psi=R(r)\Theta(\theta) \!$$

This makes the Laplacian,
 * $$\nabla^2\psi=\frac{\Theta}{r}\frac{\partial}{\partial r}(r\frac{\partial R}{\partial r})+\frac{R}{r^2}\frac{\partial^2 \Theta}{\partial \theta^2}$$

Multiplying by $$r^2 \!$$ and dividing by $$R\Theta \!$$,
 * $$\nabla^2\psi=\frac{r}{R}\frac{d}{dr}(r\frac{dR}{dr})+\frac{1}{\Theta}\frac{d^2\Theta}{d\theta^2}$$

Or,

 * $$\alpha(r)+\beta(\theta)=0 \!$$
 * $$\Rightarrow \alpha(r)=-\beta(\theta)=k^2$$

The two separated equations are then,

 * 1) $$r^2\frac{d^2R}{dr^2}+r\frac{dR}{dr}-k^2R=0$$
 * 2) $$\frac{d^2\Theta}{d\theta^2}+k^2\Theta=0$$

For (1), we can use the trial solution, $$R=r^{\lambda} \!$$, $$R'=\lambda r^{\lambda-1} \!$$, $$R''=\lambda(\lambda-1)r^{\lambda-2} \!$$

Plugging in yields,
 * $$\lambda(\lambda-1)+\lambda-k^2=0 \!$$
 * $$\Rightarrow \lambda^2=k^2 \!$$

For (2), we can use the trial solution $$\Theta = \exp\sigma \!$$, $$\Theta'=\sigma\exp\sigma \!$$, $$\Theta''=\sigma^2\exp\sigma \!$$

Plugging in yields,
 * $$\sigma^2+\lambda^2=0 \! $$
 * $$\Rightarrow \sigma^2=-\lambda^2 \!$$

Problem 9
p. 33-3 For $$f=\sum_i g_i$$ 1) show if {gi} is odd then f is odd 2) show if {gi} is even then f is even

A continuous function f can be expressed as $$f(x)=\sum_{n=0}^\infty A_nP_n(x)$$

We have observed that when n is EVEN $$P_n$$ is EVEN and when n is ODD $$P_n$$ is ODD

If we say $$g_i=A_nP_n$$

Then $$g_i$$ will be EVEN for all EVEN values of n.

And $$g_i$$ will be ODD for all ODD values of n.

Problem 10
p. 33-3 Show $$P_{2k}(x)$$ is even for k=0,1,2,... Show $$P_{2k+1}(x)$$ is odd for k=0,1,2,... Use equation (6) from p.31-3

For $$P_{2k}(x)$$ 'n' will always be an even number for all values of 'k' i.e. k=0,1,2,3,4,5......; n=0,2,4,6,8,10....

For $$P_{2k+1}(x)$$ 'n' will always be an odd number for all values of 'k' i.e. k=0,1,2,3,4,5......; n=1,3,5,7,9,11....

From the equation $$P_n(x)=\sum_{i=0}^{[n/2]}(-1)^i{(2n-2i)!x^{n-2i}\over 2^ni!(n-i)!(n-2i)!}$$ where [n/2] is only the integer part of the quotient i.e. [3/2]=1.5=1

$$\ x^{n-2i}$$ will always be an even number for all even values of 'n'.

$$\ x^{n-2i}$$ will always be an odd number for all odd values of 'n'.

Problem 11
p. 33-4 For $$q(x)=\sum_{i=0}^4c_ix^i$$ where $$c_0=3,\ c_1=10,\ c_2=15,\ c_3=-1,\ c_4=5$$

Find $$(a_i)$$ such that $$q=\sum_{i=0}^4a_iP_i$$

Plot $$q=\sum_ic_ix^i\ and\ q=\sum_ia_iP_i$$

$$q(x)=\sum_{i=0}^4c_ix^i$$ can be shown as a column matrix:

$$\begin{bmatrix} c_0x^0\\c_1x^1\\c_2x^2\\c_3x^3\\c_4x^4 \end{bmatrix}=\begin{bmatrix} 3x^0\\10x^1\\15x^2\\-1x^3\\5x^4 \end{bmatrix}$$

$$q(x)=\sum_{i=0}^4a_iP_i=\begin{bmatrix} a_0&a_1&a_2&a_3&a_4 \end{bmatrix}\begin{bmatrix} P_0\\P_1\\P_2\\P_3\\P_4 \end{bmatrix}$$

Setting $$q(x)=\sum_{i=0}^4c_ix^i=\sum_{i=0}^4a_iP_i(x)$$ yields

$$\begin{bmatrix} a_0&a_1&a_2&a_3&a_4 \end{bmatrix}\begin{bmatrix} P_0\\P_1\\P_2\\P_3\\P_4 \end{bmatrix}=\begin{bmatrix} 3x^0\\10x^1\\15x^2\\-1x^3\\5x^4 \end{bmatrix}$$

$$a_0P_0=3x^0 \Rightarrow a_0={3/P_0}$$

$$a_1P_1=10x^1 \Rightarrow a_1={10x/P_1}$$

$$a_2P_1=15x^2 \Rightarrow a_2={15x^2/P_2}$$

$$a_3P_3=-1x^3 \Rightarrow a_3={-x^3/P_3}$$

$$a_4P_4=5x^4 \Rightarrow a_4={5x^4/P_4}$$

From lecture p.31-3

$$P_0(x)=1$$

$$P_1(x)=x$$

$$P_2(x)=1/2(3x^2-1)$$

$$P_3(x)=1/2(5x^3-3x)$$

$$P_4(x)=25/8x^4-15/4x^2+3/8$$

 $$a_0=3$$

$$a_1={10x\over x}=10$$

$$a_2={15x^2\over 1/2(3x^2-1)}$$

$$a_3={-x^3\over 1/2(5x^3-3x)}$$

$$a_4={5x^4\over 25/8x^4-15/4x^2+3/8}$$

Fig. 1 Fig. 2

Problem 12
Complete homework from p.22-2 if not done on previous HW#4

See HW#4 Problem 5

Problem 13
p. 34-2 Given the b.c.$$\psi(1,\theta)=f(\theta)=T_0cos\theta$$

1) Without calculation find the properties of An 2)  Compute three (3) non-zero coefficients for An using the following methods: 2a)Analytically (integration table or direct integration) use either $$\theta$$ or $$\mu$$.  2b)Numerically using Gaus-Legendre accurately within 5%

1) Since $$f \!$$ is even, $$A_n \!$$ will be zero when $$P_n \!$$ is odd or $$n=2k+1 \!$$

2a) The first 3 non-zero coefficients are $$ n=0,2,4 \!$$ calculated as follows:

 * $$A_n=\frac{2n+1}{2}\int_{-1}^{1}f(x)P_n(x)dx$$ with, $$f(x)=T_0\sqrt{1-x^2} \!$$.

To facilitate integration the equation can be converted to the variable $$\theta \!$$ by,

 * $$x=\sin{\theta} \!$$, $$dx=\cos{\theta}d\theta \!$$
 * $$f(\theta)=T_0\cos{\theta} \!$$
 * $$x=-1, \theta = -\frac{\pi}{2} \!$$, $$x=1, \theta = \frac{\pi}{2} \!$$

The new integral is then,

 * $$A_n=\frac{2n+1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}T_0\cos{\theta}P_n(\sin{\theta})\cos\theta d\theta \!$$

For $$n=0,2,4 \!$$ the equations are,

 * $$A_0=\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}T_0\cos^2{\theta}P_0(\sin{\theta}) d\theta \!$$
 * $$A_2=\frac{5}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}T_0\cos^2{\theta}P_2(\sin{\theta}) d\theta \!$$
 * $$A_4=\frac{9}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}T_0\cos^2{\theta}P_4(\sin{\theta}) d\theta \!$$

And

 * $$P_0(\sin{\theta})=1 \!$$
 * $$P_2(\sin{\theta})=\frac{1}{2}(3\sin^2{\theta}-1) \!$$
 * $$P_4(\sin{\theta})=\frac{35}{8}\sin^4{\theta}-\frac{15}{4}\sin^2{\theta}+\frac{3}{8} \!$$

The identity to aid in integration is,

 * $$\int\cos^n{\theta}d\theta = \frac{\cos^{n-1}{\theta}\sin{\theta}}{n}+\frac{n-1}{n}\int\cos^{n-2}{\theta}d\theta \!$$

Making sure to convert any sine terms into cosine, the results are as follows,

 * $$A_0=\frac{\pi T_0}{4} \!$$
 * $$A_2=\frac{-5\pi T_0}{32} \!$$
 * $$A_4=\frac{-9\pi T_0}{256} \!$$

2b) Numerically solving for An can be done using the Gauss-Legendre quadrature. An integral from [-1,1] can be approximated by,

 * $$\sum_{j=1}^{m}w_jg(x_j) \!$$ where $$x_j \!$$ are the roots of the Legendre polynomial $$P_m \!$$
 * $$w_j = \frac{2}{(m+1)P'_m(x_j)P_{m+1}(x_j)}\!$$

For our problem,

 * $$g(x_j)=T_0\sqrt{1-x^2_j} P_n(x_j) \!$$

Therefore, $$A_n \!$$ can be written,

 * $$A_n = \frac{2n+1}{2}T_0\sum_{j=1}^{m}w_j\sqrt{1-x_j^2}P_n(x_j) \!$$

Writing out each function for n=0,2,4,

 * $$A_0 = \frac{1}{2}T_0\sum_{j=1}^{m}\frac{2\sqrt{1-x_j^2}P_0(x_j)}{(m+1)P'_m(x_j)P_{m+1}(x_j)} \!$$
 * $$A_2 = \frac{5}{2}T_0\sum_{j=1}^{m}\frac{2\sqrt{1-x_j^2}P_2(x_j)}{(m+1)P'_m(x_j)P_{m+1}(x_j)} \!$$
 * $$A_4 = \frac{9}{2}T_0\sum_{j=1}^{m}\frac{2\sqrt{1-x_j^2}P_4(x_j)}{(m+1)P'_m(x_j)P_{m+1}(x_j)} \!$$

Choosing $$m=10 \!$$ gives us the accuracy we need (5%) in determining each $$A_n\!$$,

 * $$A_0 = 0.7858 T_0 \!$$
 * $$A_2 =-0.4890 T_0 \!$$
 * $$A_4 =-0.1067 T_0 \!$$

Problem 14
p. 35-3 Verify the table for the Gaus-Legendre quadrature in wikepedia, analytic expression of {xj} and {wj} for j=1,...,n and n=1,...,5. You must first verify the expression for Pn for n=5,6. Evaluate numerically {xj} and {wj} and compare results with Abramovitz & Stegun, Table 25.4 p.916.

First we will verify $$P_n\ when\ n=5,6$$

$$P_n(x)=\sum_{i=0}^{[n/2]}(-1)^i{(2n-2i)!x^{n-2i}\over 2^ni!(n-i)!(n-2i)!}$$

When n=5

$$P_{5(i=0)}(x)=(-1)^0{10!x^5\over 2^5(0!)(5!)(5!)}={(1)3628800x^5\over 460800}={63\over 8}x^5$$

$$P_{5(i=1)}(x)=(-1)^1{8!x^3\over 2^5(1!)(4!)(3!)}={(-1)40320x^3\over 4608}={-70\over 8}x^3$$

$$P_{5(i=2)}(x)=(-1)^2{6!x\over 2^5(2!)(3!)(1!)}={(1)720x\over 384}={15\over 8}x$$

$$P_5(x)=P_{5(i=0)}+P_{5(i=1)}+P_{5(i=2)}={63\over 8}x^5-{70\over 8}x^3+{15\over 8}x$$  $$P_5(x)={1\over 8}(63x^5-70x^3+15x)$$ When n=6

$$P_{6(i=0)}(x)=(-1)^0{12!x^6\over 2^6(0!)(6!)(6!)}={(1)479001600x^6\over 33177600}={231\over 16}x^6$$

$$P_{6(i=1)}(x)=(-1)^1{10!x^3\over 2^6(1!)(5!)(4!)}={(-1)3628800x^4\over 184320}={-315\over 16}x^4$$

$$P_{6(i=2)}(x)=(-1)^2{8!x^2\over 2^6(2!)(4!)(2!)}={(1)40320x^2\over 6144}={105\over 16}x^2$$

$$P_{6(i=3)}(x)=(-1)^3{6!x^0\over 2^6(3!)(3!)(0!)}={(-1)720\over 2304}=-5$$

$$P_6(x)=P_{6(i=0)}+P_{6(i=1)}+P_{6(i=2)}+P_{6(i=3)}={231\over 16}x^6-{315\over 16}x^4+{105\over 16}x^2-5$$  $$P_6(x)={1\over 16}(231x^6-315x^4+105x^2-5)$$

$$P_1,\ P_2,\ P_3,\ P_4$$ have been shown earlier in problem #6 of HW#6

Solve for the roots of $$(x_i)\ for\ n=1,..,5$$

$$P_1(x)=x\Rightarrow roots\ x_1=0$$

$$P_2(x)=1/2(3x^2-1)\Rightarrow x^2=1/3\Rightarrow roots\ x_2=+/-\sqrt {1/3}$$

$$P_3(x)=1/2(5x^3-3x)\Rightarrow (5/2)x^3=(3/2)x\Rightarrow x^2=3/5\Rightarrow roots\ x_3=0,\ x_3=+/-\sqrt{3/5}$$

$$P_4(x)=1/8(35x^4-30x^2+3)\Rightarrow roots\ x_4=+/-\sqrt {(3-2\sqrt 6/5)\over 7},\ x_4=+/-\sqrt {(3+2\sqrt 6/5)\over 7}$$

$$P_5(x)=1/8(63x^5-70x^3-15x)\Rightarrow roots\ x_5=0, x_5=+/-1/3\sqrt {5-2\sqrt 10/7},\ x_5=+/-1/3\sqrt {5+2\sqrt 10/7}$$

Solve for {wj} for j=1,...n and n=1,...5

where $$w_j={-2\over (n+1)P'_n(x_j)P_{n+1}(x_j)}$$

n=1

$$P_1=x;\ P'_1=1;\ P_2=1/2(3x_j^2-1)$$  $$w_1={-2\over (2)(1)(-1/2)}=2$$

n=2

$$P_2=1/2(3x_j^2-1);\ P'_2=3x_j;\ P_3=1/2(5x_j^3-3x)$$  $$w_{2+}={-2\over (3)(3)\sqrt{1/3}(1/2)[5\sqrt{1/3}^3-3\sqrt{1/3}]}={-2\over 9(1/3)^{1/2}[5/2(1/3)^{3/2}-3/2(1/3)^{1/2}]}={-2\over -2}=1$$

$$w_{2-}={-2\over (3)(3)(-\sqrt{1/3})(1/2)[5(-\sqrt{1/3}^3)-3(-\sqrt{1/3})]}={-2\over -9(1/3)^{1/2}[5/2(-1/3)^{3/2}-3/2(-1/3)^{1/2}]}={-2\over -2}=1$$

n=3

$$P_3=1/2(5x_j^3-3x);\ P'_3=15/2x_j^2-3/2;\ P_4=1/8(35x^4-30x^2+3)$$  $$w_{3-0}={-2\over (4)[(15/2)(0)-3/2](1/8)[35(0)-30(0)+3]}={-2\over (4)(-3/2)(3/8)}={-2\over -9/4}=8/9$$

$$w_{3+}={-2\over (4)[(15/2)\sqrt{3/5}^2-3/2](1/8)[35\sqrt{3/5}^4-30\sqrt{3/5}^2+3]}={-2\over (4)(3)(-18/5)}={-2\over -3.6}=5/9$$

$$w_{3-}={-2\over (4)[(15/2)(-\sqrt{3/5}^2)-3/2](1/8)[35(-\sqrt{3/5}^4)-30(-\sqrt{3/5}^2)+3]}={-2\over (4)(3)(-18/5)}={-2\over -3.6}=5/9$$

n=4

$$P_4=1/8(35x^4-30x^2+3);\ P'_4=1/8(140x^3-60x);\ P_5=1/8(63x^5-70x^3+15x)$$  $$w_{4+-}={-2\over (5)\Bigg[1/8\Bigg(140\Bigg(\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^3-60\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)\Bigg]\Bigg[1/8\Bigg(63\Bigg(\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^5-70\Bigg(\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^3+15\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)\Bigg]}={18+\sqrt 30\over 36}$$

$$w_{4--}={-2\over (5)\Bigg[1/8\Bigg(140\Bigg(-\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^3-60-\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)\Bigg]\Bigg[1/8\Bigg(63\Bigg(-\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^5-70\Bigg(-\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)^3+15\Bigg(-\sqrt {(3-2\sqrt 6/5)\over 7}\Bigg)\Bigg)\Bigg]}={18+\sqrt 30\over 36}$$

$$w_{4++}={-2\over (5)\Bigg[1/8\Bigg(140\Bigg(\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^3-60\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)\Bigg]\Bigg[1/8\Bigg(63\Bigg(\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^5-70\Bigg(\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^3+15\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)\Bigg]}={18-\sqrt 30\over 36}$$

$$w_{4-+}={-2\over (5)\Bigg[1/8\Bigg(140\Bigg(-\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^3-60-\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)\Bigg]\Bigg[1/8\Bigg(63\Bigg(-\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^5-70\Bigg(-\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)^3+15\Bigg(-\sqrt {(3+2\sqrt 6/5)\over 7}\Bigg)\Bigg)\Bigg]}={18-\sqrt 30\over 36}$$

n=5

$$P_5=1/8(63x^5-70x^3+15x);\ P'_5=1/8(315x^4-210x^2+15);\ P_6=1/16(231x^6-315x^4+105x^2-5)$$

 $$w_{5-0}={-2\over (6)\Big[1/8\Big(315(0)-210(0)+15\Big)\Big]\Big[1/16\Big(231(0)-315(0)-5\Big)\Big]}={-2\over (6)(15/8)(-5/16)}=128/225$$

$$w_{5+-}={-2\over (6)\Bigg[1/8\Bigg(315\Bigg(1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^4-210\Bigg(1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^2+15\Bigg)\Bigg]\Bigg[1/16\Bigg(231\Bigg(1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^6-315\Bigg(1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^4+105\Bigg(1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^2-5\Bigg)\Bigg]}={332+13\sqrt70\over 900}$$

$$w_{5--}={-2\over (6)\Bigg[1/8\Bigg(315\Bigg(-1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^4-210\Bigg(-1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^2+15\Bigg)\Bigg]\Bigg[1/16\Bigg(231\Bigg(-1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^6-315\Bigg(-1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^4+105\Bigg(-1/3\sqrt{5-2\sqrt{10/7}}\Bigg)^2-5\Bigg)\Bigg]}={332+13\sqrt70\over 900}$$

$$w_{5++}={-2\over (6)\Bigg[1/8\Bigg(315\Bigg(1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^4-210\Bigg(1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^2+15\Bigg)\Bigg]\Bigg[1/16\Bigg(231\Bigg(1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^6-315\Bigg(1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^4+105\Bigg(1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^2-5\Bigg)\Bigg]}={332-13\sqrt70\over 900}$$

$$w_{5-+}={-2\over (6)\Bigg[1/8\Bigg(315\Bigg(-1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^4-210\Bigg(-1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^2+15\Bigg)\Bigg]\Bigg[1/16\Bigg(231\Bigg(-1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^6-315\Bigg(-1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^4+105\Bigg(-1/3\sqrt{5+2\sqrt{10/7}}\Bigg)^2-5\Bigg)\Bigg]}={332-13\sqrt70\over 900}$$

Numerical values of {xj} and {wj}

$$\begin{bmatrix} {} & x_i & w_i \\ n=2 &

0.577350269189626 & 1.000000000000000 \\ n=3 & 0.000000000000000 & 0.888888888888889 \\ n=3 & 0.774596669241483 & 0.555555555555556 \\ n=4 & 0.339981043584856 & 0.652145154862546 \\ n=4 & 0.861136311594053 & 0.347854845137454\\ n=5 & 0.000000000000000 & 0.568888888888889 \\ n=5 & 0.538469310105683 & 0.478628670499366 \\ n=5 & 0.906179845938664 & 0.236926885056189 \end{bmatrix}$$

Problem 15
p. 36-4 show:

$$Q_0(x)=\frac{1}{2}log\Bigg({1+x\over 1-x}\Bigg)=tanh^{-1}(x)$$

If

 * $$\frac{1}{2}log\Bigg({\frac{1+x}{1-x}}\Bigg)=\tanh^{-1}{x}$$

then

 * $$\tanh{\Bigg(\tanh^{-1}x\Bigg)} = x $$

or

 * $$\tanh{\Bigg(\frac{1}{2}\log\Bigg({\frac{1+x}{1-x}}\Bigg) \Bigg)}=x $$

where

 * $$\tanh{x}=\frac{e^{2x}-1}{e^{2x}+1} $$

Plugging in gives

 * $$\frac{e^{\log{\frac{1+x}{1-x}}}-1}{e^{\log{\frac{1+x}{1-x}}}+1} \Rightarrow \frac{\frac{1+x}{1-x}-1}{\frac{1+x}{1-x}+1} $$

 * $$\frac{1+x-1+x}{1+x+1-x} = \frac{2x}{2}$$

 * $$\Rightarrow x \!$$

Contributing Team Members
--Joe Gaddone 20:59, 16 November 2009 (UTC) Kumanchik 21:13, 16 November 2009 (UTC) --Matt Walker 1400, 18 November 2009 (UTC) Egm6321.f09.Team2.sungsik 20:08, 18 November 2009 (UTC)

title: User:Egm6321.f09.Team2/HW6


text:
Differential equations/Assignment 1

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics and other disciplines.

title: Differential equations/Assignment 1


text:
Talk:Communities of Practice

The normal wikipedia article is way more complete than this. Mecha-ant 23:51, 9 November 2009 (UTC)

title: Talk:Communities of Practice


text:
User:Egm6321.f09.TA/Exam1grading

Exam 1 point breakdown

Consider the general nonlinear 2nd-order ODE of the form $$\sqrt{x}y''+\frac{2+x^3}{\sqrt{x}}y'+4x^{3/2}y=0$$. (1.1)  Question 1

Consider the general nonlinear second order ODE of the form $$F(x,y,y',y'')=0$$. (1.2) Let $$\phi(x,y,y')$$ denote an exact function (first integral) for the above ODE. Show that for this to be exact, it must have the following form $$F(x,y,y',y)=f(x,y,y')y+g(x,y,y')=0$$,  (1.3) where $$f$$ and $$g$$ are two functions of $$ (x,y,y')$$. Express $$f,g$$ in terms of $$\phi$$. The above result is the first condition for exactness for nonlinear 2nd-order ODEs. 10 points for the total derivative of $$\phi$$: $$\frac{d\phi}{dx}=\phi_x+p\phi_y +y''\phi_p$$.

10 points for partitioning the derivative as$$f=\phi_p$$ and $$ g=\phi_x +p\phi_y$$.  Question 2

Next, the second condition for exactness of nonlinear 2nd-order ODEs is given as follows. Let $$p:= y'$$, we have $$f_{xx}+2pf_{xy}+p^2f_{yy}=g_{xp}+pg_{yp}-g_y$$ (1.4) $$f_{xp}+pf_{yp}+2f_y=g_{pp} $$ (1.5)

Derive the above exactness condition by eliminating $$\phi$$. To this end, differentiate the results in the previous question with respect to $$x,y,$$ and $$ p$$ to eliminate $$\phi$$. 15 points for equation 1.4(5 points for each partial, $$\phi_{xp},\phi_{yp},\phi_{xy}$$) 5 points for equation 1.5  Question 3

Verify wheter (1.1) satisfy (a) The first condition for exactness.10 points (b) The second condition for exactness.10 points  Question 4

Find a first integral $$\phi$$ whether or not (1.1) is exact. Hint: If (1.1) is not exact, use the integrating factor method with the integrating factor $$ h(x,y)=x^a y^b$$, (1.6)

5 points for constructing the $$f,g$$ equations. 5 points for $$a=3/2$$. 5 points for $$b=0$$. 5 points for solving $$\phi=x^2p+x^4y$$.  Question 5

Find the solution $$y(x)$$ for (1.1)

5 points partial credit were awarded for using any solution technique (e.g., trial solution), particularly in cases where the $$\phi$$ derived in question 4 was incorrect and/or cumbersome.

10 points for solving for $$y(x)$$

10 points for any solution containing $$y=Ce^{-\frac{1}{3}x^3}$$. Some students solved $$\phi=x^2p+x^4y=0$$(separable) and some solved $$\phi=x^2p+x^4y=C$$ (more complicated). Both were accepted. --Egm6321.f09.TA 01:34, 10 November 2009 (UTC)

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