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University of Canberra/Things that don't work/Barriers to online learning

Converted from: Barriers to online learning at the University of Canberra - these recommendations were not included in the Review of Online and Blended Learning. This link is PW protected.

Initial recommendations by James Neill
Feel free to add suggested recommendations for the Online and Blended Learning Review. NOTE: This is a working space to inform the review, not a final record.

 * 1) Documentation Review: A vast amount of UC documentation, proformas etc. is paper-based. A systematic audit could reveal many areas of potential efficiency gain by converting to electronic systems (e.g., print-room orders are currently hard-copy based). This would better enable the institution to undertake efficient and effective online and blended teaching. With the trend towards outsourcing of many functions to off-site locations, developing an efficient electronic organisation is paramount.
 * 2) IP Policy Review: Online and blended learning could be significantly facilitated by adopting a more innovative Intellectual Property (IP) policy. The current IP policy is well overdue for review. UC could benefit from a systematic review of other institutions' more innovative licensing policies. In practical terms, teachers and researchers should be able select at least from the suite of license options (e.g., creative commons licenses) for created content. The current practice is that either the IP policy is ignored and other licenses are applied anyway - or innovation is restrained by those who follow the current policy.

Barriers to online learning
A list of roadblocks/barriers in trying to offer online learning @ UC (in other words, a list of things which arguably warrant consideration for change). See also University of Canberra/Things that don't work.

Hardware

 * 1) Only four lecture theatres (2B7, 2B9, 2B11 and 14B1) are equipped for video and audio recording.
 * 2) Few staff have webcams - and even fewer use them e.g., for video introductions to students, etc.

File formats

 * 1) Open file formats are not systematically used/embraced.

Unavailable

 * 1) VOIP
 * 2) Desktop SMS
 * 3) * A corporate solution may be to implement SMS via Moodle e.g.,
 * 4) ** http://moodle.org/mod/forum/discuss.php?d=35798
 * 5) ** http://moodle.org/mod/forum/discuss.php?d=109344
 * 6) Desktop video-conferencing

Underutilised

 * 1) Wiki: UC has a functional, supported corporate wiki (confluence / ucspace) which is underutilised in part due to the lack of corporate roll-out e.g., no staff or student training has been offered for its use, even though it's been available for several years.

Forms etc. to be transitioned to electronic format / electronic submission
Much of the proforma documentation at UC remains in 20th century hard-copy format, requiring to staff to be on campus, walk back and forth between printers, hand-deliver documents (or use internal mail), etc. Documentation which is in electronic submission format is done in an ad hoc manner by various areas e.g., some use PDF (e.g., exam booking). A university-wide review of proforma documentation and implementation of a more systematic solution could considerably increase efficiency, flexibility, and readiness for online and blended learning. Examples:

 * 1) AV lecture recording booking form is hard-copy submitted, not electronic.
 * 2) Current policy (which one?) indicates that a hard copy of the unit outline is to be distributed to students (?check this)
 * 3) Assignment coversheet is currently designed for hard-copy assignments
 * 4) Final marks require printing out and manual signing (rather than electronic signing) - which also raises the broader issue that UC doesn't have a corporate system for electronic signatures
 * 5) Amendment to Examiner's Return is hard-copy - could be electronic
 * 6) ADEs currently send hard-copy letters to students to advise of alleged plagiarism
 * 7) Referee forms for student applications are hard-copy and inefficient for staff to complete.
 * 8) For more examples, see student forms
 * 9) Assignment extension requests are generally done via hard-copy, discipline-specific proformas
 * 10) Print room order forms are hard-copy and max. 10MB attachments can be emailed to the printroom


text:
Gospel of John (CBS)/Week 2

Congratulations on surviving the first week! Tests will be graded shortly and I will post the results when I am done. This week we begin Unit 2: The Book of Signs. This unit will continue through Week 5, when there will be another Unit Test.

Assignments

 * 1) Read John 1:19-2:11
 * 2) Write a well developed response that expands upon this week's topic.
 * 3) Begin working on the Study Guide
 * 4) Check test score feedback (optional, but highly advisable!!)

Discussion
This week's discussion will be held at TIME on DATE in the AWF virtual universe, Agape World at the AWF School facility. There is a section on the talk page where you can post topics and questions for us to go over. If you are unable to make it, your questions can also be posted in a separate section to the talk page, where I will do my best to answer them shortly.

Handouts

 * Study Guide -60 questions, due before Unit 2 Test in Week 5

Lesson
The Book of Signs is the second part of the Gospel of John. It is in this part of the Gospel that Jesus reveals himself as the Son of God and Savior of Mankind. Before we go into this, we need to address a key difference of John from the Synoptic Gospels of Matthew, Mark, and Luke. Unlike the Synoptics, the author of this Gospel chose to ignore Jesus's early life. Instead, the author jumps right into the beginning of Jesus's ministry. We don't know why this is the case, and overall it doesn't really matter. The big focus is the Gospel that Jesus brought, not minutiae about his life.

In his earthly ministry, Jesus revealed himself as Messiah gradually. He was considered a great teacher, or Rabbi, and taught the Good News in the temple before he began to reveal himself. Many times he was asked if he was the Messiah, and he answered in ways that his questioners were unable to understand. If Jesus had simply claimed to be Messiah, no one would have cause to believe him. In order to lay out his claim, he had to gradually reveal himself through miracles, signs that fulfilled prophecies about his Coming.

These miracles form the signs that the Book of Signs is named for. Ultimately, this chapter is all about miracles, signs and wonders that proved that Jesus really was who he said he is. In the reading from this week, he starts out by turning water into wine, a fairly minor feat. By the time we are done studying this book, he will have moved onto the second biggest miracle he performed, the resurrection of a human being!

As Jesus performed these miracles, he caused an uproar among Jews. Each miracle builds upon the last one to demonstrate another facet of the Messiah, shaking up Jewish society. At the end, there could be no doubt that Jesus was the Messiah foretold in prophecy.

Chapter 1
''John's public testimony concerning Christ. (19-28) Other testimonies of John concerning Christ. (29-36) Andrew and another disciple follow Jesus. (37-42) Philip and Nathanael called. (43-51) '' Verses 19-28 John disowns himself to be the Christ, who was now expected and waited for. He came in the spirit and power of Elias, but he was not the person of Elias. John was not that Prophet whom Moses said the Lord would raise up to them of their brethren, like unto him. He was not such a prophet as they expected, who would rescue them from the Romans. He gave such an account of himself, as might excite and awaken them to hearken to him. He baptized the people with water as a profession of repentance, and as an outward sign of the spiritual blessings to be conferred on them by the Messiah, who was in the midst of them, though they knew him not, and to whom he was unworthy to render the meanest service.

Verses 29-36 John saw Jesus coming to him, and pointed him out as the Lamb of God. The paschal lamb, in the shedding and sprinkling of its blood, the roasting and eating of its flesh, and all the other circumstances of the ordinance, represented the salvation of sinners by faith in Christ. And the lambs sacrificed every morning and evening, can only refer to Christ slain as a sacrifice to redeem us to God by his blood. John came as a preacher of repentance, yet he told his followers that they were to look for the pardon of their sins to Jesus only, and to his death. It agrees with God's glory to pardon all who depend on the atoning sacrifice of Christ. He takes away the sin of the world; purchases pardon for all that repent and believe the gospel. This encourages our faith; if Christ takes away the sin of the world, then why not my sin? He bore sin for us, and so bears it from us. God could have taken away sin, by taking away the sinner, as he took away the sin of the old world; but here is a way of doing away sin, yet sparing the sinner, by making his Son sin, that is, a sin-offering, for us. See Jesus taking away sin, and let that cause hatred of sin, and resolutions against it. Let us not hold that fast, which the Lamb of God came to take away. To confirm his testimony concerning Christ, John declares the appearance at his baptism, in which God himself bore witness to him. He saw and bare record that he is the Son of God. This is the end and object of John's testimony, that Jesus was the promised Messiah. John took every opportunity that offered to lead people to Christ.

Verses 37-42 The strongest and most prevailing argument with an awakened soul to follow Christ, is, that it is he only who takes away sin. Whatever communion there is between our souls and Christ, it is he who begins the discourse. He asked, What seek ye? The question Jesus put to them, we should all put to ourselves when we begin to follow Him, What do we design and desire? In following Christ, do we seek the favour of God and eternal life? He invites them to come without delay. Now is the ( 2 Corinthians 6:2 ) is, wherever it be. We ought to labour for the spiritual welfare of those related to us, and seek to bring them to Him. Those who come to Christ, must come with a fixed resolution to be firm and constant to him, like a stone, solid and stedfast; and it is by his grace that they are so.

Verses 43-51 See the nature of true Christianity, it is following Jesus; devoting ourselves to him, and treading in his steps. Observe the objection Nathanael made. All who desire to profit by the word of God, must beware of prejudices against places, or denominations of men. They should examine for themselves, and they will sometimes find good where they looked for none. Many people are kept from the ways of religion by the unreasonable prejudices they conceive. The best way to remove false notions of religion, is to make trial of it. In Nathanael there was no guile. His profession was not hypocritical. He was not a dissembler, nor dishonest; he was a sound character, a really upright, godly man. Christ knows what men are indeed. Does He know us? Let us desire to know him. Let us seek and pray to be Israelites indeed, in whom is no guile; truly Christians, approved of Christ himself. Some things weak, imperfect, and sinful, are found in all, but hypocrisy belongs not to a believer's character. Jesus witnessed what passed when Nathanael was under the fig-tree. Probably he was then in fervent prayer, seeking direction as to the Hope and Consolation of Israel, where no human eye observed him. This showed him that our Lord knew the secrets of his heart. Through Christ we commune with, and benefit by the holy angels; and things in heaven and things on earth are reconciled and united together.knew the secrets of his heart. Through Christ we commune with, and benefit by the holy angels; and things in heaven and things on earth are reconciled and united together.

Chapter 2
''The miracle at Cana. (1-11)'' Verses 1-11 It is very desirable when there is a marriage, to have Christ own and bless it. Those that would have Christ with them at their marriage, must invite him by prayer, and he will come. While in this world we sometimes find ourselves in straits, even when we think ourselves in fulness. There was want at a marriage feast. Those who are come to care for the things of the world, must look for trouble, and count upon disappointment. In our addresses to Christ, we must humbly spread our case before him, and then refer ourselves to him to do as he pleases. In Christ's reply to his mother there was no disrespect. He used the same word when speaking to her with affection from the cross; yet it is a standing testimony against the idolatry of after-ages, in giving undue honours to his mother. His hour is come when we know not what to do. Delays of mercy are not denials of prayer. Those that expect Christ's favours, must observe his orders with ready obedience. The way of duty is the way to mercy; and Christ's methods must not be objected against. The beginning of Moses' miracles was turning water into blood, ( Exodus 7:20 ) ; the beginning of Christ's miracles was turning water into wine; which may remind us of the difference between the law of Moses and the gospel of Christ. He showed that he improves creature-comforts to all true believers, and make them comforts indeed. And Christ's works are all for use. Has he turned thy water into wine, given thee knowledge and grace? it is to profit withal; therefore draw out now, and use it. It was the best wine. Christ's works commend themselves even to those who know not their Author. What was produced by miracles, always was the best in its kind. Though Christ hereby allows a right use of wine, he does not in the least do away his own caution, which is, that our hearts be not at any time overcharged with surfeiting and drunkenness, ( Luke 21:34 ). Though we need not scruple to feast with our friends on proper occasions, yet every social interview should be so conducted, that we might invite the Redeemer to join with us, if he were now on earth; and all levity, luxury, and excess offend him.

Unit Test
The Unit Test will be similar to the last Unit Test. There will be 30 free response questions drawn from information covered in readings and lessons for this unit. Many of the questions may be reworded study questions, so the study guide is your friend :). While it isn't going to happen until Week 5, you will receive the best score if you begin studying now. Attempt to complete at least 15 questions from the study guide this week, especially those which relate to this weeks topic.

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

title: Gospel of John (CBS)/Week 2


text:
Gospel of John (CBS)/Week 3

We're now in the third week of this course and continuing in our studies of the Book of Signs. You will notice there are no handouts this week, as the Study Guide for this unit was given last week. That is the only major difference this week.

Assignments

 * 1) Read John 2:12-4:54
 * 2) Write a well developed response that expands upon this week's topic.
 * 3) Continue working on the Study Guide

Discussion
This week's discussion will be held at TIME on DATE in the AWF virtual universe, Agape World at the AWF School facility. There is a section on the talk page where you can post topics and questions for us to go over. If you are unable to make it, your questions can also be posted in a separate section to the talk page, where I will do my best to answer them shortly.

Lesson
When Jesus came, the Jews had developed many institutions. These institutions were part of the Law and upheld the Law. They were no longer necessary, had become corrupted in some instances, and needed to be replaced. Through his ministry, Jesus demonstrated the failings and fallacies inherent in these institutions. People were no longer worshiping God, but rather tradition and institution. It was time for a radical cleansing and purification preparatory to the greatest miracle of all time.

The reading for this week starts off with one major example of a defective institution, the Temple merchants. Passover was an important time for Jews, and was one of the most important feasts with certain very specific requirements for sacrifices. Many people were able to meet the sacrifice requirements, but some did not have the necessary lambs, turtledoves or oxen. They were still required to make the sacrifice, which created a void to be fulfilled. Merchants were more than happy to meet this need, and over time the Temple became filled with stalls selling necessary animals and changing currency. This institution was created to fulfill the Law, but had several problems mainly relating to the intent of the sacrifices. Wiktionary defines a sacrifice as, "to give away (something valuable) to get at least a possibility to gain something else of value (such as self-respect, trust, love, freedom, prosperity), or to avoid an even greater loss". value is inherent in the sacrifice, or it isn't a sacrifice. God wanted sacrifices as an act of atonement. Through a sacrifice that had value, it would directly impact the person making it, leading them to the realization of the effect of sin. When you can buy the animal for sacrifice, where is the value in it? It loses its meaning and purpose. You aren't sacrificing something meaningful out of respect for God, you are making the sacrifice because its socially acceptable and expected. This worship of institution and tradition removed God from the Temple, a problem that could not be allowed to stand. To add insult to injury, there were money changers too! The Temple wasn't a place to go to worship the Lord, it was a marketplace! A house of business, not a house of worship.

When Jesus saw what the Jews had done to His Father's House, he was deeply grieved and angered. He went through and drove out the filth, seeking to restore the Temple to some vestige of its original purpose. The Jews had strayed so far from God that this perplexed them. What authority did this mere common man have to come into the Temple and propose a radical change? They challenged him and demanded a miraculous sign to justify his action. This opportunity allowed Jesus the chance to make a radical change in the location of the Temple. "Destroy this temple and I will rebuild it in three days". The Jews did not understand him, because they assumed he was thinking about the Temple building. In reality, he was talking about his body and foretelling how long it would be before his resurrection. God no longer needed the Temple building, and history tells us that it was destroyed shortly thereafter. Jesus replaced the Old Testament Temple with a New Testament Temple, the personal Temple. Through His sacrifice, the Temple resides within us. Now that our sin has been removed, God's presence can now dwell in man!

The Jews were too caught up in the law to understand this, making Jesus an enigma to them. They had never seen anyone like him. He healed, he taught, and he brought a new truth and hope to Israel. Some were angered by his healings on the Sabbath, others were frightened about the power he was amassing. The one thing in common all Jews shared is they were unsure of how to react to this man who acted like the Messiah, yet spoke in ways they could not understand.

Chapter 2
''Christ casts the buyers and sellers out of the temple. (12-22) Many believe in Christ. (23-25)'' Verses 12-22 The first public work in which we find Christ engaged, was driving from the temple the traders whom the covetous priests and rulers encouraged to make a market-place of its courts. Those now make God's house a house of merchandise, whose minds are filled with cares about worldly business when attending religious exercises, or who perform Divine offices for love of gain. Christ, having thus cleansed the temple, gave a sign to those who demanded it, to prove his authority for so doing. He foretells his death by the Jews' malice, Destroy ye this temple; I will permit you to destroy it. He foretells his resurrection by his own power; In three days I will raise it up. Christ took again his own life. Men mistake by understanding that according to the letter, which the Scripture speaks by way of figure. When Jesus was risen from the dead, his disciples remembered he has said this. It helps much in understanding the Divine word, to observe the fulfilling of the Scriptures.

Verses 23-25 Our Lord knew all men, their nature, dispositions, affections, designs, so as we do not know any man, not even ourselves. He knows his crafty enemies, and all their secret projects; his false friends, and their true characters. He knows who are truly his, knows their uprightness, and knows their weaknesses. We know what is done by men; Christ knows what is in them, he tries the heart. Beware of a dead faith, or a formal profession: carnal, empty professors are not to be trusted, and however men impose on others or themselves, they cannot impose on the heart-searching God.

Chapter 3
''Christ's discourse with Nicodemus. (1-21) The baptism of John of Christ John's testimony. (22-36)''

Verses 1-8 Nicodemus was afraid, or ashamed to be seen with Christ, therefore came in the night. When religion is out of fashion, there are many Nicodemites. But though he came by night, Jesus bid him welcome, and hereby taught us to encourage good beginnings, although weak. And though now he came by night, yet afterward he owned Christ publicly. He did not talk with Christ about state affairs, though he was a ruler, but about the concerns of his own soul and its salvation, and went at once to them. Our Saviour spoke of the necessity and nature of regeneration or the new birth, and at once directed Nicodemus to the source of holiness of the heart. Birth is the beginning of life; to be born again, is to begin to live anew, as those who have lived much amiss, or to little purpose. We must have a new nature, new principles, new affections, new aims. By our first birth we were corrupt, shapen in sin; therefore we must be made new creatures. No stronger expression could have been chosen to signify a great and most remarkable change of state and character. We must be entirely different from what we were before, as that which begins to be at any time, is not, and cannot be the same with that which was before. This new birth is from heaven, ch. 1:13, and its tendency is to heaven. It is a great change made in the heart of a sinner, by the power of the Holy Spirit. It means that something is done in us, and for us, which we cannot do for ourselves. Something is wrong, whereby such a life begins as shall last for ever. We cannot otherwise expect any benefit by Christ; it is necessary to our happiness here and hereafter. What Christ speak, Nicodemus misunderstood, as if there had been no other way of regenerating and new-moulding an immortal soul, than by new-framing the body. But he acknowledged his ignorance, which shows a desire to be better informed. It is then further explained by the Lord Jesus. He shows the Author of this blessed change. It is not wrought by any wisdom or power of our own, but by the power of the blessed Spirit. We are shapen in iniquity, which makes it necessary that our nature be changed. We are not to marvel at this; for, when we consider the holiness of God, the depravity of our nature, and the happiness set before us, we shall not think it strange that so much stress is laid upon this. The regenerating work of the Holy Spirit is compared to water. It is also probable that Christ had reference to the ordinance of baptism. Not that all those, and those only, that are baptized, are saved; but without that new birth which is wrought by the Spirit, and signified by baptism, none shall be subjects of the kingdom of heaven. The same word signifies both the wind and the Spirit. The wind bloweth where it listeth for us; God directs it. The Spirit sends his influences where, and when, on whom, and in what measure and degree, he pleases. Though the causes are hidden, the effects are plain, when the soul is brought to mourn for sin, and to breathe after Christ. Christ's stating of the doctrine and the necessity of regeneration, it should seem, made it not clearer to Nicodemus. Thus the things of the Spirit of God are foolishness to the natural man. Many think that cannot be proved, which they cannot believe. Christ's discourse of gospel truths, ver. ( 11-13 ), shows the folly of those who make these things strange unto them; and it recommends us to search them out. Jesus Christ is every way able to reveal the will of God to us; for he came down from heaven, and yet is in heaven. We have here a notice of Christ's two distinct natures in one person, so that while he is the Son of man, yet he is in heaven. God is the "HE THAT IS," and heaven is the dwelling-place of his holiness. The knowledge of this must be from above, and can be received by faith alone. Jesus Christ came to save us by healing us, as the children of Israel, stung with fiery serpents, were cured and lived by looking up to the brazen serpent, ( Numbers 21:6-9 ). In this observe the deadly and destructive nature of sin. Ask awakened consciences, ask damned sinners, they will tell you, that how charming soever the allurements of sin may be, at the last it bites like a serpent. See the powerful remedy against this fatal malady. Christ is plainly set forth to us in the gospel. He whom we offended is our Peace, and the way of applying for a cure is by believing. If any so far slight either their disease by sin, or the method of cure by Christ, as not to receive Christ upon his own terms, their ruin is upon their own heads. He has said, Look and be saved, look and live; lift up the eyes of your faith to Christ crucified. And until we have grace to do this, we shall not be cured, but still are wounded with the stings of Satan, and in a dying state. Jesus Christ came to save us by pardoning us, that we might not die by the sentence of the law. Here is gospel, good news indeed. Here is God's love in giving his Son for the world. God so loved the world; so really, so richly. Behold and wonder, that the great God should love such a worthless world! Here, also, is the great gospel duty, to believe in Jesus Christ. God having given him to be our Prophet, Priest, and King, we must give up ourselves to be ruled, and taught, and saved by him. And here is the great gospel benefit, that whoever believes in Christ, shall not perish, but shall have everlasting life. God was in Christ reconciling the world to himself, and so saving it. It could not be saved, but through him; there is no salvation in any other. From all this is shown the happiness of true believers; he that believeth in Christ is not condemned. Though he has been a great sinner, yet he is not dealt with according to what his sins deserve. How great is the sin of unbelievers! God sent One to save us, that was dearest to himself; and shall he not be dearest to us? How great is the misery of unbelievers! they are condemned already; which speaks a certain condemnation; a present condemnation. The wrath of God now fastens upon them; and their own hearts condemn them. There is also a condemnation grounded on their former guilt; they are open to the law for all their sins; because they are not by faith interested in the gospel pardon. Unbelief is a sin against the remedy. It springs from the enmity of the heart of man to God, from love of sin in some form. Read also the doom of those that would not know Christ. Sinful works are works of darkness. The wicked world keep as far from this light as they can, lest their deeds should be reproved. Christ is hated, because sin is loved. If they had not hated saving knowledge, they would not sit down contentedly in condemning ignorance. On the other hand, renewed hearts bid this light welcome. A good man acts truly and sincerely in all he does. He desires to know what the will of God is, and to do it, though against his own worldly interest. A change in his whole character and conduct has taken place. The love of God is shed abroad in his heart by the Holy Ghost, and is become the commanding principle of his actions. So long as he continues under a load of unforgiven guilt, there can be little else than slavish fear of God; but when his doubts are done away, when he sees the righteous ground whereon this forgiveness is built, he rests on it as his own, and is united to God by unfeigned love. Our works are good when the will of God is the rule of them, and the glory of God the end of them; when they are done in his strength, and for his sake; to him, and not to men. Regeneration, or the new birth, is a subject to which the world is very averse; it is, however, the grand concern, in comparison with which every thing else is but trifling. What does it signify though we have food to eat in plenty, and variety of raiment to put on, if we are not born again? if after a few mornings and evenings spent in unthinking mirth, carnal pleasure, and riot, we die in our sins, and lie down in sorrow? What does it signify though we are well able to act our parts in life, in every other respect, if at last we hear from the Supreme Judge, "Depart from me, I know you not, ye workers of iniquity?"

Verses 22-36 John was fully satisfied with the place and work assigned him; but Jesus came on a more important work. He also knew that Jesus would increase in honour and influence, for of his government and peace there would be no end, while he himself would be less followed. John knew that Jesus came from heaven as the Son of God, while he was a sinful, mortal man, who could only speak about the more plain subjects of religion. The words of Jesus were the words of God; he had the Spirit, not by measure, as the prophets, but in all fulness. Everlasting life could only be had by faith in Him, and might be thus obtained; whereas all those, who believe not in the Son of God, cannot partake of salvation, but the wrath of God for ever rests upon them.

Chapter 4
''Christ's departure into Galilee. (1-3) His discourse with the Samaritan woman. (4-26) The effects of Christ's conversation with the woman of Samaria. (27-42) Christ heals the nobleman's son. (43-54) ''

Verses 1-3 Jesus applied himself more to preaching, which was the more ( 1 Corinthians 1:17 ) his disciples, by employing them to baptize. He teaches us that the benefit of sacraments depends not on the hand that administers them.

Verses 4-26 There was great hatred between the Samaritans and the Jews. Christ's road from Judea to Galilee lay through Samaria. We should not go into places of temptation but when we needs must; and then must not dwell in them, but hasten through them. We have here our Lord Jesus under the common fatigue of travellers. Thus we see that he was truly a man. Toil came in with sin; therefore Christ, having made himself a curse for us, submitted to it. Also, he was a poor man, and went all his journeys on foot. Being wearied, he sat thus on the well; he had no couch to rest upon. He sat thus, as people wearied with travelling sit. Surely, we ought readily to submit to be like the Son of God in such things as these. Christ asked a woman for water. She was surprised because he did not show the anger of his own nation against the Samaritans. Moderate men of all sides are men wondered at. Christ took the occasion to teach her Divine things: he converted this woman, by showing her ignorance and sinfulness, and her need of a Saviour. By this living water is meant the Spirit. Under this comparison the blessing of the Messiah had been promised in the Old Testament. The graces of the Spirit, and his comforts, satisfy the thirsting soul, that knows its own nature and necessity. What Jesus spake figuratively, she took literally. Christ shows that the water of Jacob's well yielded a very short satisfaction. Of whatever waters of comfort we drink, we shall thirst again. But whoever partakes of the Spirit of grace, and the comforts of the gospel, shall never want that which will abundantly satisfy his soul. Carnal hearts look no higher than carnal ends. Give it me, saith she, not that I may have everlasting life, which Christ proposed, but that I come not hither to draw. The carnal mind is very ingenious in shifting off convictions, and keeping them from fastening. But how closely our Lord Jesus brings home the conviction to her conscience! He severely reproved her present state of life. The woman acknowledged Christ to be a prophet. The power of his word in searching the heart, and convincing the conscience of secret things, is a proof of Divine authority. It should cool our contests, to think that the things we are striving about are passing away. The object of worship will continue still the same, God, as a Father; but an end shall be put to all differences about the place of worship. Reason teaches us to consult decency and convenience in the places of our worship; but religion gives no preference to one place above another, in respect of holiness and approval with God. The Jews were certainly in the right. Those who by the Scriptures have obtained some knowledge of God, know whom they worship. The word of salvation was of the Jews. It came to other nations through them. Christ justly preferred the Jewish worship before the Samaritan, yet here he speaks of the former as soon to be done away. God was about to be revealed as the Father of all believers in every nation. The spirit or the soul of man, as influenced by the Holy Spirit, must worship God, and have communion with him. Spiritual affections, as shown in fervent prayers, supplications, and thanksgivings, form the worship of an upright heart, in which God delights and is glorified. The woman was disposed to leave the matter undecided, till the coming of the Messiah. But Christ told her, I that speak to thee, am He. She was an alien and a hostile Samaritan, merely speaking to her was thought to disgrace our Lord Jesus. Yet to this woman did our Lord reveal himself more fully than as yet he had done to any of his disciples. No past sins can bar our acceptance with him, if we humble ourselves before him, believing in him as the Christ, the Saviour of the world.

Verses 27-42 The disciples wondered that Christ talked thus with a Samaritan. Yet they knew it was for some good reason, and for some good end. Thus when particular difficulties occur in the word and providence of God, it is good to satisfy ourselves that all is well that Jesus Christ says and does. Two things affected the woman. The extent of his knowledge. Christ knows all the thoughts, words, and actions, of all the children of men. And the power of his word. He told her secret sins with power. She fastened upon that part of Christ's discourse, many would think she would have been most shy of repeating; but the knowledge of Christ, into which we are led by conviction of sin, is most likely to be sound and saving. They came to him: those who would know Christ, must meet him where he records his name. Our Master has left us an example, that we may learn to do the will of God as he did; with diligence, as those that make a business of it; with delight and pleasure in it. Christ compares his work to harvest-work. The harvest is appointed and looked for before it comes; so was the gospel. Harvest-time is busy time; all must be then at work. Harvest-time is a short time, and harvest-work must be done then, or not at all; so the time of the gospel is a season, which if once past, cannot be recalled. God sometimes uses very weak and unlikely instruments for beginning and carrying on a good work. Our Saviour, by teaching one poor woman, spread knowledge to a whole town. Blessed are those who are not offended at Christ. Those taught of God, are truly desirous to learn more. It adds much to the praise of our love to Christ and his word, if it conquers prejudices. Their faith grew. In the matter of it: they believed him to be the Saviour, not only of the Jews but of the world. In the certainty of it: we know that this is indeed the Christ. And in the ground of it, for we have heard him ourselves.

Verses 43-54 The father was a nobleman, yet the son was sick. Honours and titles are no security from sickness and death. The greatest men must go themselves to God, must become beggars. The nobleman did not stop from his request till he prevailed. But at first he discovered the weakness of his faith in the power of Christ. It is hard to persuade ourselves that distance of time and place, are no hinderance to the knowledge, mercy, and power of our Lord Jesus. Christ gave an answer of peace. Christ's saying that the soul lives, makes it alive. The father went his way, which showed the sincerity of his faith. Being satisfied, he did not hurry home that night, but returned as one easy in his own mind. His servants met him with the news of the child's recovery. Good news will meet those that hope in God's word. Diligent comparing the works of Jesus with his word, will confirm our faith. And the bringing the cure to the family brought salvation to it. Thus an experience of the power of one word of Christ, may settle the authority of Christ in the soul. The whole family believed likewise. The miracle made Jesus dear to them. The knowledge of Christ still spreads through families, and men find health and salvation to their souls.

Unit Test
The Unit Test will be similar to the last Unit Test. There will be 30 free response questions drawn from information covered in readings and lessons for this unit. Many of the questions may be reworded study questions, so the study guide is your friend :). Attempt to complete at least 15 questions from the study guide this week, especially those which relate to this weeks topic.

title: Gospel of John (CBS)/Week 3


text:
Gospel of John (CBS)/Week 4

We're coming up on the halfway point of this course and continuing in our studies of the Book of Signs. Keep up the good work! We're also heading towards a Unit Test next week, so you should begin to review for it if you haven't already.

Assignments

 * 1) Read John 5:1-10:42
 * 2) Write a well developed response that expands upon this week's topic.
 * 3) Continue working on the Study Guide
 * 4) Review material for next week's unit test

Discussion
This week's discussion will be held at TIME on DATE in the AWF virtual universe, Agape World at the AWF School facility. There is a section on the talk page where you can post topics and questions for us to go over. If you are unable to make it, your questions can also be posted in a separate section to the talk page, where I will do my best to answer them shortly.

Chapter 5
''The cure at the pool of Bethesda. (1-9) The Jews' displeasure. (10-16) Christ reproves the Jews. (17-23) Christ's discourse. (24-47)''

Verses 1-9 We are all by nature impotent folk in spiritual things, blind, halt, and withered; but full provision is made for our cure, if we attend to it. An angel went down, and troubled the water; and what disease soever it was, this water cured it, but only he that first stepped in had benefit. This teaches us to be careful, that we let not a season slip which may never return. The man had lost the use of his limbs thirty-eight years. Shall we, who perhaps for many years have scarcely known what it has been to be a day sick, complain of one wearisome night, when many others, better than we, have scarcely known what it has been to be a day well? Christ singled this one out from the rest. Those long in affliction, may comfort themselves that God keeps account how long. Observe, this man speaks of the unkindness of those about him, without any peevish reflections. As we should be thankful, so we should be patient. Our Lord Jesus cures him, though he neither asked nor thought of it. Arise, and walk. God's command, Turn and live; Make ye a new heart; no more supposes power in us without the grace of God, his distinguishing grace, than this command supposed such power in the impotent man: it was by the power of Christ, and he must have all the glory. What a joyful surprise to the poor cripple, to find himself of a sudden so easy, so strong, so able to help himself! The proof of spiritual cure, is our rising and walking. Has Christ healed our spiritual diseases, let us go wherever he sends us, and take up whatever he lays upon us; and walk before him.

Verses 10-16 Those eased of the punishment of sin, are in danger of returning to sin, when the terror and restraint are over, unless Divine grace dries up the fountain. The misery believers are made whole from, warns us to sin no more, having felt the smart of sin. This is the voice of every providence, Go, and sin no more. Christ saw it necessary to give this caution; for it is common for people, when sick, to promise much; when newly recovered, to perform only something; but after awhile to forget all. Christ spoke of the wrath to come, which is beyond compare worse than the many hours, nay, weeks and years of pain, some wicked men have to suffer in consequence of their unlawful indulgences. And if such afflictions are severe, how dreadful will be the everlasting punishment of the wicked!

Verses 17-23 The Divine power of the miracle proved Jesus to be the Son of God, and he declared that he worked with, and like unto his Father, as he saw good. These ancient enemies of Christ understood him, and became more violent, charging him not only with sabbath-breaking, but blasphemy, in calling God his own Father, and making himself equal with God. But all things now, and at the final judgment, are committed to the Son, purposely that all men might honour the Son, as they honour the Father; and every one who does not thus honour the Son, whatever he may think or pretend, does not honour the Father who sent him.

Verses 24-29 Our Lord declared his authority and character, as the Messiah. The time was come when the dead should hear his voice, as the Son of God, and live. Our Lord first refers to his raising those who were dead in sin, to newness of life, by the power of the Spirit, and then to his raising the dead in their graves. The office of Judge of all men, can only be exercised by one who has all knowledge, and almighty power. May we believe His testimony; thus our faith and hope will be in God, and we shall not come into condemnation. And may His voice reach the hearts of those dead in sin; that they may do works meet for repentance, and prepare for the solemn day.

Verses 30-38 Our Lord returns to his declaration of the entire agreement between the Father and the Son, and declared himself the Son of God. He had higher testimony than that of John; his works bore witness to all he had said. But the Divine word had no abiding-place in their hearts, as they refused to believe in Him whom the Father had sent, according to his ancient promises. The voice of God, accompanied by the power of the Holy Ghost, thus made effectual to the conversion of sinners, still proclaims that this is the beloved Son, in whom the Father is well pleased. But when the hearts of men are full of pride, ambition, and the love of the world, there is no room for the word of God to abide in them.

Verses 39-44 The Jews considered that eternal life was revealed to them in their Scriptures, and that they had it, because they had the word of God in their hands. Jesus urged them to search those Scriptures with more diligence and attention. "Ye do search the Scriptures," and ye do well to do so. They did indeed search the Scriptures, but it was with a view to their own glory. It is possible for men to be very studious in the letter of the Scriptures, yet to be strangers to its power. Or, "Search the Scriptures," and so it was spoken to them in the nature of an appeal. Ye profess to receive and believe the Scripture, let that be the judge. It is spoken to us as advising or commanding all Christians to search the Scriptures. Not only read them, and hear them, but search them; which denotes diligence in examining and studying them. We must search the Scriptures for heaven as our great end; For in them ye think ye have eternal life. We must search the Scriptures for Christ, as the new and living Way, that leads to this end. To this testimony Christ adds reproofs of their unbelief and wickedness; their neglect of him and his doctrine. Also he reproves their want of the love of God. But there is life with Jesus Christ for poor souls. Many who make a great profession of religion, yet show they want the love of God, by their neglect of Christ and contempt of his commandments. It is the love of God in us, the love that is a living, active principle in the heart, which God will accept. They slighted and undervalued Christ, because they admired and overvalued themselves. How can those believe, who make the praise and applause of men their idol! When Christ and his followers are men wondered at, how can those believe, the utmost of whose ambition is to make a fair show in the flesh!

Verses 45-47 Many trust in some form of doctrines or some parties, who no more enter into the real meaning of those doctrines, or the views of the persons whose names they bear, than the Jews did into those of Moses. Let us search and pray over the Scriptures, as intent on finding eternal life; let us observe how Christ is the great subject of them, and daily apply to him for the life he bestows.

Chapter 6
''Five thousand miraculously fed. (1-14) Jesus walks on the sea. (15-21) He directs to spiritual food. (22-27) His discourse with the multitude. (28-65) Many of disciples go back. (66-71)''

Verses 1-14 John relates the miracle of feeding the multitude, for its reference to the following discourse. Observe the effect this miracle had upon the people. Even the common Jews expected the Messiah to come into the world, and to be a great Prophet. The Pharisees despised them as not knowing the law; but they knew most of Him who is the end of the law. Yet men may acknowledge Christ as that Prophet, and still turn a deaf ear to him.

Verses 15-21 Here were Christ's disciples in the way of duty, and Christ was praying for them; yet they were in distress. There may be perils and afflictions of this present time, where there is an interest in Christ. Clouds and darkness often surround the children of the light and of the day. They see Jesus walking on the sea. Even the approaches of comfort and deliverance often are so mistaken, as to become the occasions of fear. Nothing is more powerful to convince sinners than that word, "I am Jesus whom thou persecutest;" nothing more powerful to comfort saints than this, "I am Jesus whom thou lovest." If we have received Christ Jesus the Lord, though the night be dark, and the wind high, yet we may comfort ourselves, we shall be at the shore before long.

Verses 22-27 Instead of answering the inquiry how he came there, Jesus blamed their asking. The utmost earnestness should be employed in seeking salvation, in the use of appointed means; yet it is to be sought only as the gift of the Son of man. Him the Father has sealed, proved to be God. He declared the Son of man to be the Son of God with power.

Verses 28-35 Constant exercise of faith in Christ, is the most important and difficult part of the obedience required from us, as sinners seeking salvation. When by his grace we are enabled to live a life of faith in the Son of God, holy tempers follow, and acceptable services may be done. God, even his Father, who gave their fathers that food from heaven to support their natural lives, now gave them the true Bread for the salvation of their souls. Coming to Jesus, and believing on him, signify the same. Christ shows that he is the true Bread; he is to the soul what bread is to the body, nourishes and supports the spiritual life. He is the Bread of God. Bread which the Father gives, which he has made to be the food of our souls. Bread nourishes only by the powers of a living body; but Christ is himself living Bread, and nourishes by his own power. The doctrine of Christ crucified is now as strengthening and comforting to a believer as ever it was. He is the Bread which came down from heaven. It denotes the Divinity of Christ's person and his authority; also, the Divine origin of all the good which flows to us through him. May we with understanding and earnestness say, Lord, evermore give us this Bread.

Verses 36-46 The discovery of their guilt, danger, and remedy, by the teaching of the Holy Spirit, makes men willing and glad to come, and to give up every thing which hinders applying to him for salvation. The Father's will is, that not one of those who were given to the Son, should be rejected or lost by him. No one will come, till Divine grace has subdued, and in part changed his heart; therefore no one who comes will ever be cast out. The gospel finds none willing to be saved in the humbling, holy manner, made known therein; but God draws with his word and the Holy Ghost; and man's duty is to hear and learn; that is to say, to receive the grace offered, and consent to the promise. None had seen the Father but his beloved Son; and the Jews must expect to be taught by his inward power upon their minds, and by his word, and the ministers whom he sent among them.

Verses 47-51 The advantage of the manna was small, it only referred to this life; but the living Bread is so excellent, that the man who feedeth on it shall never die. This bread is Christ's human nature, which he took to present to the Father, as a sacrifice for the sins of the world; to purchase all things pertaining to life and godliness, for sinners of every nation, who repent and believe in him.

Verses 52-59 The flesh and blood of the Son of man, denote the Redeemer in the nature of man; Christ and him crucified, and the redemption wrought out by him, with all the precious benefits of redemption; pardon of sin, acceptance with God, the way to the throne of grace, the promises of the covenant, and eternal life. These are called the flesh and blood of Christ, because they are purchased by the breaking his body, and the shedding of his blood. Also, because they are meat and drink to our souls. Eating this flesh and drinking this blood mean believing in Christ. We partake of Christ and his benefits by faith. The soul that rightly knows its state and wants, finds whatever can calm the conscience, and promote true holiness, in the redeemer, God manifest in the flesh. Meditating upon the cross of Christ gives life to our repentance, love, and gratitude. We live by him, as our bodies live by our food. We live by him, as the members by the head, the branches by the root: because he lives we shall live also.

Verses 60-65 The human nature of Christ had not before been in heaven, but being God and man, that wondrous Person was truly said to have come down from heaven. The Messiah's kingdom was not of this world; and they were to understand by faith, what he had said of a spiritual living upon him, and his fulness. As without the soul of man the flesh is of no value, so without the quickening Spirit of God all forms of religion are dead and worthless. He who made this provision for our souls, alone can teach us these things, and draw us unto Christ, that we may live by faith in him. Let us apply to Christ, thankful that it is declared that every one who is willing to come unto him shall be made welcome.

Verses 66-71 When we admit into our minds hard thoughts of the words and works of Jesus, we enter into temptation, which, if the Lord in mercy prevent not, will end in drawing back. The corrupt and wicked heart of man often makes that an occasion for offence, which is matter of the greatest comfort. Our Lord had, in the foregoing discourse, promised eternal life to his followers; the disciples fastened on that plain saying, and resolved to cleave to him, when others fastened on hard sayings, and forsook him. Christ's doctrine is the word of eternal life, therefore we must live and die by it. If we forsake Christ, we forsake our own mercies. They believed that this Jesus was the Messiah promised to their fathers, the Son of the living God. When we are tempted to backslide or turn away, it is good to remember first principles, and to keep to them. And let us ever remember our Lord's searching question; Shall we go away and forsake our Redeemer? To whom can we go? He alone can give salvation by the forgiveness of sins. And this alone brings confidence, comfort, and joy, and bids fear and despondency flee away. It gains the only solid happiness in this world, and opens a way to the happiness of the next.

Chapter 7
''Christ goes to the feast of tabernacles. (1-13) His discourse at the feast. (14-39) The people dispute concerning Christ. (40-53)''

Verses 1-13 The brethren or kinsmen of Jesus were disgusted, when they found there was no prospect of worldly advantages from him. Ungodly men sometimes undertake to counsel those employed in the work of God; but they only advise what appears likely to promote present advantages. The people differed about his doctrine and miracles, while those who favoured him, dared not openly to avow their sentiments. Those who count the preachers of the gospel to be deceivers, speak out, while many who favour them, fear to get reproach by avowing regard for them.

Verses 14-24 Every faithful minister may humbly adopt Christ's words. His doctrine is not his own finding out, but is from God's word, through the teaching of his Spirit. And amidst the disputes which disturb the world, if any man, of any nation, seeks to do the will of God, he shall know whether the doctrine is of God, or whether men speak of themselves. Only those who hate the truth shall be given up to errors which will be fatal. Surely it was as agreeable to the design of the sabbath to restore health to the afflicted, as to administer an outward rite. Jesus told them to decide on his conduct according to the spiritual import of the Divine law. We must not judge concerning any by their outward appearance, but by their worth, and by the gifts and graces of God's Spirit in them.

Verses 25-30 Christ proclaimed aloud, that they were in error in their thoughts about his origin. He was sent of God, who showed himself true to his promises. This declaration, that they knew not God, with his claim to peculiar knowledge, provoked the hearers; and they sought to take him, but God can tie men's hands, though he does not turn their hearts.

Verses 31-36 The discourses of Jesus convinced many that he was the Messiah; but they had not courage to own it. It is comfort to those who are in the world, but not of it, and therefore are hated by it and weary of it, that they shall not be in it always, that they shall not be in it long. Our days being evil, it is well they are few. The days of life and of grace do not last long; and sinners, when in misery, will be glad of the help they now despise. Men dispute about such sayings, but the event will explain them.

Verses 37-39 On the last day of the feast of tabernacles, the Jews drew water and poured it out before the Lord. It is supposed that Christ alluded to this. If any man desires to be truly and for ever happy, let him apply to Christ, and be ruled by him. This thirst means strong desires after spiritual blessings, which nothing else can satisfy; so the sanctifying and comforting influences of the Holy Spirit, were intended by the waters which Jesus called on them to come to Him and drink. The comfort flows plentifully and constantly as a river; strong as a stream to bear down the opposition of doubts and fears. There is a fulness in Christ, of grace for grace. The Spirit dwelling and working in believers, is as a fountain of living, running water, out of which plentiful streams flow, cooling and cleansing as water. The miraculous gifts of the Holy Spirit we do not expect, but for his more common and more valuable influences we may apply. These streams have flowed from our glorified Redeemer, down to this age, and to the remote corners of the earth. May we be anxious to make them known to others.

Verses 40-53 The malice of Christ's enemies is always against reason, and sometimes the staying of it cannot be accounted for. Never any man spake with that wisdom, and power, and grace, that convincing clearness, and that sweetness, wherewith Christ spake. Alas, that many, who are for a time restrained, and who speak highly of the word of Jesus, speedily lose their convictions, and go on in their sins! People are foolishly swayed by outward motives in matters of eternal moment, are willing even to be damned for fashion's sake. As the wisdom of God often chooses things which men despise, so the folly of men commonly despises those whom God has chosen. The Lord brings forward his weak and timid disciples, and sometimes uses them to defeat the designs of his enemies.

Chapter 8
''The Pharisees and the adulteress. (1-11) Christ's discourse with the Pharisees. (12-59)''

Verses 1-11 Christ neither found fault with the law, nor excused the prisoner's guilt; nor did he countenance the pretended zeal of the Pharisees. Those are self-condemned who judge others, and yet do the same thing. All who are any way called to blame the faults of others, are especially concerned to look to themselves, and keep themselves pure. In this matter Christ attended to the great work about which he came into the world, that was, to bring sinners to repentance; not to destroy, but to save. He aimed to bring, not only the accused to repentance, by showing her his mercy, but the prosecutors also, by showing them their sins; they thought to insnare him, he sought to convince and convert them. He declined to meddle with the magistrate's office. Many crimes merit far more severe punishment than they meet with; but we should not leave our own work, to take that upon ourselves to which we are not called. When Christ sent her away, it was with this caution, Go, and sin no more. Those who help to save the life of a criminal, should help to save the soul with the same caution. Those are truly happy, whom Christ does not condemn. Christ's favour to us in the forgiveness of past sins should prevail with us, Go then, and sin no more.

Verses 12-16 Christ is the Light of the world. God is light, and Christ is the image of the invisible God. One sun enlightens the whole world; so does one Christ, and there needs no more. What a dark dungeon would the world be without the sun! So would it be without Jesus, by whom light came into the world. Those who follow Christ shall not walk in darkness. They shall not be left without the truths which are necessary to keep them from destroying error, and the directions in the way of duty, necessary to keep them from condemning sin.

Verses 17-20 If we knew Christ better, we should know the Father better. Those become vain in their imaginations concerning God, who will not learn of Christ. Those who know not his glory and grace, know not the Father that sent him. The time of our departure out of the world, depends upon God. Our enemies cannot hasten it any sooner, nor can our friends delay it any longer, than the time appointed of the Father. Every true believer can look up and say with pleasure, My times are in thy hand, and better there than in my own. To all God's purposes there is a time.

Verses 21-29 Those that live in unbelief, are for ever undone, if they die in unbelief. The Jews belonged to this present evil world, but Jesus was of a heavenly and Divine nature, so that his doctrine, kingdom, and blessings, would not suit their taste. But the curse of the law is done away to all that submit to the grace of the gospel. Nothing but the doctrine of Christ's grace will be an argument powerful enough, and none but the Spirit of Christ's grace will be an agent powerful enough, to turn us from sin to God; and that Spirit is given, and that doctrine is given, to work upon those only who believe in Christ. Some say, Who is this Jesus? They allow him to have been a Prophet, an excellent Teacher, and even more than a creature; but cannot acknowledge him as over all, God blessed for evermore. Will not this suffice? Jesus here answers the question. Is this to honour him as the Father? Does this admit his being the Light of the world, and the Life of men, one with the Father? All shall know by their conversion, or in their condemnation, that he always spake and did what pleased the Father, even when he claimed the highest honours to himself.

Verses 30-36 Such power attended our Lord's words, that many were convinced, and professed to believe in him. He encouraged them to attend his teaching, rely on his promises, and obey his commands, notwithstanding all temptations to evil. Thus doing, they would be his disciples truly; and by the teaching of his word and Spirit, they would learn where their hope and strength lay. Christ spoke of spiritual liberty; but carnal hearts feel no other grievances than those that molest the body, and distress their worldly affairs. Talk to them of their liberty and property, tell them of waste committed upon their lands, or damage done to their houses, and they understand you very well; but speak of the bondage of sin, captivity to Satan, and liberty by Christ; tell of wrong done to their precious souls, and the hazard of their eternal welfare, then you bring strange things to their ears. Jesus plainly reminded them, that the man who practised any sin, was, in fact, a slave to that sin, which was the case with most of them. Christ in the gospel offers us freedom, he has power to do this, and those whom Christ makes free are really so. But often we see persons disputing about liberty of every kind, while they are slaves to some sinful lust.

Verses 37-40 Our Lord opposed the proud and vain confidence of these Jews, showing that their descent from Abraham could not profit those of a contrary spirit to him. Where the word of God has no place, no good is to be expected; room is left there for all wickedness. A sick person who turns from his physician, and will take neither remedies nor food, is past hope of recovery. The truth both heals and nourishes the hearts of those who receive it. The truth taught by philosophers has not this power and effect, but only the truth of God. Those who claim the privileges of Abraham, must do Abraham's works; must be strangers and sojourners in this world; keep up the worship of God in their families, and always walk before God.

Verses 41-47 Satan prompts men to excesses by which they murder themselves and others, while what he puts into the mind tends to ruin men's souls. He is the great promoter of falsehood of every kind. He is a liar, all his temptations are carried on by his calling evil good, and good evil, and promising freedom in sin. He is the author of all lies; whom liars resemble and obey, with whom all liars shall have their portion for ever. The special lusts of the devil are spiritual wickedness, the lusts of the mind, and corrupt reasonings, pride and envy, wrath and malice, enmity to good, and enticing others to evil. By the truth, here understand the revealed will of God as to the salvation of men by Jesus Christ, the truth Christ was now preaching, and which the Jews opposed.

Verses 48-53 Observe Christ's disregard of the applause of men. those who are dead to the praises of men can bear their contempt. God will seek the honour of all who do not seek their own. In these verses we have the doctrine of the everlasting happiness of believers. We have the character of a believer; he is one that keeps the sayings of the Lord Jesus. And the privilege of a believer; he shall by no means see death for ever. Though now they cannot avoid seeing death, and tasting it also, yet they shall shortly be where it will be no more forever, ( Exodus 14:13 ).

Verses 54-59 Christ and all that are his, depend upon God for honour. Men may be able to dispute about God, yet may not know him. Such as know not God, and obey not the gospel of Christ, are put ( 2 Thessalonians 1:8 ) earnestly desire to know more of him. Those who discern the dawn of the light of the Sun of Righteousness, wish to see his rising. "Before Abraham was, I AM." This speaks Abraham a creature, and our Lord the Creator; well, therefore, might he make himself greater than Abraham. I AM, is the name of God, Ex 3:14; it speaks his self-existence; he is the First and the Last, ever the same, ( Revelation 1:8 ). Thus he was not only before Abraham, but before all worlds, ( Proverbs 8:23, John 1:1 ). As Mediator, he was the appointed Messiah, long before Abraham; the Lamb slain from the foundation of the world, ( Revelation 13:8 ). The Lord Jesus was made of God Wisdom, Righteousness, Sanctification, and Redemption, to Adam, and Abel, and all that lived and died by faith in him, before Abraham. The Jews were about to stone Jesus for blasphemy, but he withdrew; by his miraculous power he passed through them unhurt. Let us stedfastly profess what we know and believe concerning God; and if heirs of Abraham's faith, we shall rejoice in looking forward to that day when the Saviour shall appear in glory, to the confusion of his enemies, and to complete the salvation of all who believe in him.

Chapter 9
''Christ give sight to one born blind. (1-7) The account given by the blind man. (8-12) The Pharisees question the man that had been blind. (13-17) They ask concerning him. (18-23) They cast him out. (24-34) Christ's words to the man that had been blind. (35-38) He reproves the Pharisees. (39-41)''

Verses 1-7 Christ cured many who were blind by disease or accident; here he cured one born blind. Thus he showed his power to help in the most desperate cases, and the work of his grace upon the souls of sinners, which gives sight to those blind by nature. This poor man could not see Christ, but Christ saw him. And if we know or apprehend anything of Christ, it is because we were first known of him. Christ says of uncommon calamities, that they are not always to be looked on as special punishments of sin; sometimes they are for the glory of God, and to manifest his works. Our life is our day, in which it concerns us to do the work of the day. We must be busy, and not waste day-time; it will be time to rest when our day is done, for it is but a day. The approach of death should quicken us to improve all our opportunities of doing and getting good. What good we have an opportunity to do, we should do quickly. And he that will never do a good work till there is nothing to be objected against, will leave many a good work for ever undone, ( Ecclesiastes 11:4 ). Christ magnified his power, in making a blind man to see, doing that which one would think more likely to make a seeing man blind. Human reason cannot judge of the Lord's methods; he uses means and instruments that men despise. Those that would be healed by Christ must be ruled by him. He came back from the pool wondering and wondered at; he came seeing. This represents the benefits in attending on ordinances of Christ's appointment; souls go weak, and come away strengthened; go doubting, and come away satisfied; go mourning, and come away rejoicing; go blind, and come away seeing.

Verses 8-12 Those whose eyes are opened, and whose hearts are cleansed by grace, being known to be the same person, but widely different in character, live as monuments to the Redeemer's glory, and recommend his grace to all who desire the same precious salvation. It is good to observe the way and method of God's works, and they will appear the more wonderful. Apply this spiritually. In the work of grace wrought upon the soul we see the change, but we see not the hand that makes it: the way of the Spirit is like that of the wind, which thou hearest the sound of, but canst not tell whence it comes, nor whither it goes.

Verses 13-17 Christ not only worked miracles on the sabbath, but in such a manner as would give offence to the Jews, for he would not seem to yield to the scribes and Pharisees. Their zeal for mere rites consumed the substantial matters of religion; therefore Christ would not give place to them. Also, works of necessity and mercy are allowed, and the sabbath rest is to be kept, in order to the sabbath work. How many blind eyes have been opened by the preaching of the gospel on the Lord's day! how many impotent souls cured on that day! Much unrighteous and uncharitable judging comes from men's adding their own fancies to God's appointments. How perfect in wisdom and holiness was our Redeemer, when his enemies could find nothing against him, but the oft-refuted charge of breaking the sabbath! May we be enabled, by well-doing, to silence the ignorance of foolish men.

Verses 18-23 The Pharisees vainly hoped to disprove this notable miracle. They expected a Messiah, but could not bear to think that this Jesus should be he, because his precepts were all contrary to their traditions, and because they expected a Messiah in outward pomp and splendour. The fear of man brings a snare, ( Proverbs 29:25 ), and often makes people deny and disown Christ and his truths and ways, and act against their consciences. The unlearned and poor, who are simple-hearted, readily draw proper inferences from the evidences of the light of the gospel; but those whose desires are another way, though ever learning, never come to the knowledge of the truth.

Verses 24-34 As Christ's mercies are most valued by those who have felt the want of them, that have been blind, and now see; so the most powerful and lasting affections to Christ, arise from actual knowledge of him. In the work of grace in the soul, though we cannot tell when, and how, and by what steps the blessed change was wrought, yet we may take the comfort, if we can say, through grace, Whereas I was blind, now I see. I did live a worldly, sensual life, but, thanks be to God, it is now otherwise with me, ( Ephesians 5:8 ). The unbelief of those who enjoy the means of knowledge and conviction, is indeed marvellous. All who have felt the power and grace of the Lord Jesus, wonder at the wilfulness of others who reject him. He argues strongly against them, not only that Jesus was not a sinner, but that he was of God. We may each of us know by this, whether we are of God or not. What do we? What do we for God? What do we for our souls? What do we more than others?

Verses 35-38 Christ owns those who own him and his truth and ways. There is particular notice taken of such a suffer in the cause of Christ, and for the testimony of a good conscience. Our Lord Jesus graciously reveals himself to the man. Now he was made sensible what an unspeakable mercy it was, to be cured of his blindness, that he might see the Son of God. None but God is to be worshipped; so that in worshipping Jesus, he owned him to be God. All who believe in him, will worship him.

Verses 39-41 Christ came into the world to give sight to those who were spiritually blind. Also, that those who see might be made blind; that those who have a high conceit of their own wisdom, might be sealed up in ignorance. The preaching of the cross was thought to be folly by such as by carnal wisdom knew not God. Nothing fortifies men's corrupt hearts against the convictions of the word, more than the high opinion which others have of them; as if all that gained applause with men, must obtain acceptance with God. Christ silenced them. But the sin of the self-conceited and self-confident remains; they reject the gospel of grace, therefore the guilt of their sin remains unpardoned, and the power of their sin remains unbroken.

Chapter 10
''The parable of the good shepherd. (1-5) Christ the Door. (6-9) Christ the good Shepherd. (10-18) The Jews' opinion concerning Jesus. (19-21) His discourse at the feast of dedication. (22-30) The Jews attempt to stone Jesus. (31-38) He departs from Jerusalem. (39-42)''

Verses 1-5 Here is a parable or similitude, taken from the customs of the East, in the management of sheep. Men, as creatures depending on their Creator, are called the sheep of his pasture. The church of God in the world is as a sheep-fold, exposed to deceivers and persecutors. The great Shepherd of the sheep knows all that are his, guards them by his providence, guides them by his Spirit and word, and goes before them, as the Eastern shepherds went before their sheep, to set them in the way of his steps. Ministers must serve the sheep in their spiritual concerns. The Spirit of Christ will set before them an open door. The sheep of Christ will observe their Shepherd, and be cautious and shy of strangers, who would draw them from faith in him to fancies about him.

Verses 6-9 Many who hear the word of Christ, do not understand it, because they will not. But we shall find one scripture expounding another, and the blessed Spirit making known the blessed Jesus. Christ is the Door. And what greater security has the church of God than that the Lord Jesus is between it and all its enemies? He is a door open for passage and communication. Here are plain directions how to come into the fold; we must come in by Jesus Christ as the Door. By faith in him as the great Mediator between God and man. Also, we have precious promises to those that observe this direction. Christ has all that care of his church, and every believer, which a good shepherd has of his flock; and he expects the church, and every believer, to wait on him, and to keep in his pasture.

Verses 10-18 Christ is a good Shepherd; many who were not thieves, yet were careless in their duty, and by their neglect the flock was much hurt. Bad principles are the root of bad practices. The Lord Jesus knows whom he has chosen, and is sure of them; they also know whom they have trusted, and are sure of Him. See here the grace of Christ; since none could demand his life of him, he laid it down of himself for our redemption. He offered himself to be the Saviour; Lo, I come. And the necessity of our case calling for it, he offered himself for the Sacrifice. He was both the offerer and the offering, so that his laying down his life was his offering up himself. From hence it is plain, that he died in the place and stead of men; to obtain their being set free from the punishment of sin, to obtain the pardon of their sin; and that his death should obtain that pardon. Our Lord laid not his life down for his doctrine, but for his sheep.

Verses 19-21 Satan ruins many, by putting them out of conceit with the word and ordinances. Men would not be laughed out of their necessary food, yet suffer themselves thus to be laughed out of what is far more necessary. If our zeal and earnestness in the cause of Christ, especially in the blessed work of bringing his sheep into his fold, bring upon us evil names, let us not heed it, but remember our Master was thus reproached before us.

Verses 22-30 All who have any thing to say to Christ, may find him in the temple. Christ would make us to believe; we make ourselves doubt. The Jews understood his meaning, but could not form his words into a full charge against him. He described the gracious disposition and happy state of his sheep; they heard and believed his word, followed him as his faithful disciples, and none of them should perish; for the Son and the Father were one. Thus he was able to defend his sheep against all their enemies, which proves that he claimed Divine power and perfection equally with the Father.

Verses 31-38 Christ's works of power and mercy proclaim him to be over all, God blessed for evermore, that all may know and believe He is in the Father, and the Father in Him. Whom the Father sends, he sanctifies. The holy God will reward, and therefore will employ, none but such as he makes holy. The Father was in the Son, so that by Divine power he wrought his miracles; the Son was so in the Father, that he knew the whole of His mind. This we cannot by searching find out to perfection, but we may know and believe these declarations of Christ.

Verses 39-42 No weapon formed against our Lord Jesus shall prosper. He escaped, not because he was afraid to suffer, but because his hour was not come. And He who knew how to deliver himself, knows how to deliver the godly our of their temptations, and to make a way for them to escape. Persecutors may drive Christ and his gospel our of their own city or country, but they cannot drive him or it out of the world. When we know Christ by faith in our hearts, we find all that the Scripture saith of him is true.

Unit Test
The Unit Test will be similar to the last Unit Test. There will be 30 free response questions drawn from information covered in readings and lessons for this unit. Many of the questions may be reworded study questions, so the study guide is your friend :). Attempt to complete at least 15 questions from the study guide this week, especially those which relate to this weeks topic. The test will open next week and only be available through Moodle. Once this course is approved and the Moodle page is set up, a link will be placed here.

title: Gospel of John (CBS)/Week 4


text:
User:Egm6341.s10.Team4.yunseok/HW5

= Problem 5: Integration of logistic equation with $$h=2^kh $$=

Given
Refer Lecture slide 42-2 for problem statement

The logistic equation

 * {| style="width:100%" border="0" align="left"

\frac{d\overline{x}}{dt} = rx\left(1- \overline{x}\right) $$ where,
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

$$\displaystyle \overline{x} = \frac{x}{x_{max}} $$

Find
Integrate logistic equation already found $$\hat{h}$$ such that Hermit-simpson algorithm yields error($$ 10^{-6}$$)

1) Run Hermit-Simpson with $$\displaystyle h= 2^k \hat{h}$$

2) Develop and Run Forward Euler with $$\displaystyle h= 2^k \hat{h}$$

3) Develop and Run Backward Euler with $$\displaystyle h= 2^k \hat{h}$$

Solution
1) Run Hermit-Simpson algo $$\displaystyle h= 2^k \hat{h}$$

See problem 3

2) Develop and Run Forward Euler with $$\displaystyle h= 2^k \hat{h}$$

 * {| style="width:100%" border="0" align="left"

\begin{align} &\frac{d\overline{x}}{dt} = r\overline{x}\left(1- \overline{x}\right)\\ &\frac{\overline{x}_{i+1}-\overline{x}_i}{h}= r\overline{x_i}\left(1- \overline{x_i}\right)\\ &\overline{x}_{i+1}= hr\overline{x}_i\left(1- \overline{x}_i\right)+\overline{x}_i\\ &\overline{x}_{i+1}= (hr+1)\overline{x}_i-hr\overline{x}^2_i\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

MATLAB CODE:

3) Develop and Run Backward Euler with $$\displaystyle h= 2^k \hat{h}$$

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\begin{align} &\frac{d\overline{x}}{dt} = r\overline{x}\left(1- \overline{x}\right)\\ &\frac{\overline{x}_{i+1}-\overline{x}_i}{h}= r\overline{x}_{i+1}\left(1- \overline{x}_{i+1}\right)\\ &-\overline{x}_{i}= hr\overline{x}_{i+1}\left(1- \overline{x}_{i+1}\right)-\overline{x}_{i+1}\\ &=>hr\overline{x}^2_{i+1}+(1-hr)\overline{x}_{i+1}-\overline{x}_{i}= 0\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

\begin{align} &\overline{x}_{i+1}=\frac{hr-1 + \sqrt{(1-hr)^2+4hrx_i}}{2hr}\\ &\overline{x}_{i+1}=\frac{hr-1 - \sqrt{(1-hr)^2+4hrx_i}}{2hr}\\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

MATLAB CODE:

Conclusion

As the step size($$ 2^kh$$) increase, the Forwaord Euler method shows the unstable properties.

on the other hands, The Hermit simpson and Backworkd Euler shows the relatively stable properties.

= Problem 11: Circumference of Ellipse =

Given
Lecture notes [[media:Egm6341.s10.mtg42.djvu|p.42-2]]

Arc length
 * {| style="width:100%" border="0" align="left"

dl = [dx+dy]^{1/2}\,dt$$
 * $$\displaystyle
 * $$\displaystyle

 * $$ \displaystyle x = a cost $$

 * $$ \displaystyle y = b sint $$

 * }
 * }

Find

 * {| style="width:100%" border="0" align="left"

$$\displaystyle C = \int dl = a \int_{0}^{2\pi} [1-e^2cos^2t]^{1/2}\,dt$$ where, Eccentricity, $$ \displaystyle e = \Bigg[1-\frac{b^2}{a^2}\Bigg]^{1/2} $$
 * Show that,
 * Show that,
 * }
 * }

Solution

 * {| style="width:100%" border="0" align="left"

$$\displaystyle \frac {dy}{dt} = b cos t $$  and
 * $$\displaystyle \frac {dx}{dt} = -a sin t $$
 * $$\displaystyle \frac {dx}{dt} = -a sin t $$
 * }
 * }

 * {| style="width:100%" border="0" align="left"

\begin{align} dl &= [dx^2 + dy^2 ]^{1/2}\\ &= \Bigg[\bigg(\frac{dx}{dt}\bigg)^2 + \bigg(\frac{dy}{dt}\bigg)^2 \Bigg]^{1/2} dt\\ &=\Bigg[(-a sin t)^2 + (b cos t)^2\Bigg]^{1/2} dt\\ &= \Bigg[(a^2 sin^2 t) + (b^2 cos^2 t)\Bigg]^{1/2} dt\\ &= a \Bigg[sin^2 t + \left(\frac{b^2}{a^2} cos^2 t\right)\Bigg]^{1/2} dt\\ &= a \Bigg[sin^2 t + cos^2 t + \frac{b^2}{a^2} cos^2 t- cos^2 t \Bigg]^{1/2} dt\\ &= a \Bigg[1 -\left(1-\frac{b^2}{a^2}\right) cos^2 t \Bigg]^{1/2} dt\\ &= a \Bigg[1 - e^2 cos^2 t \Bigg]^{1/2} dt\\ \end{align} $$ So,
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.
 * {| style="width:100%" border="0" align="left"

 * $$\displaystyle C = \int_{0}^{2\pi} dl = a \int_{0}^{2\pi} [1-e^2cos^2t]^{1/2}\,dt$$
 * }
 * }

= Problem 11: Verification and comaprision of Problem 10 of HW5 =

Given
The given condition of No 10 problem of HW5

 * {| style="width:100%" border="0" align="left"

Refer Lecture slide 30-4 for problem statement
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

Polar form relative to focus
 * {| style="width:100%" border="0" align="left"

 * [[Image:Ellipse Polar.svg|200px]]
 * }.
 * }.
 * }.

Eq (3) from Page [[media:Egm6341.s10.mtg30.djvu|30-3]]

$$\displaystyle \ \ r(\theta) = \frac{a(1-e^2)}{1-e \, cos(\theta)} $$

In our problem, a=1

where, eccentricity $$\displaystyle e = \sin(\frac{\pi}{12}) $$

First Method to compute the Arc Length for an elliptical curve is using Eq (4) from Page [[media:Egm6341.s10.mtg30.djvu|30-3]] Since for the Ellipse $$\displaystyle \theta$$ range is from 0 to 2 $$\displaystyle\pi $$

Circumference $$\displaystyle I( \theta) = \int_0^{2\pi} dl$$

Where Eq (1) $$\displaystyle dl= \sqrt{ {r}^{2}+ {\left(  \frac{dr}{d \theta}\right)}^{2}} d\theta $$.

The second method uses the elliptic integral of the second kind which is defined as:

$$E(e^2) = \int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta\ $$    Eq(6)    in page     [[media:Egm6341.s10.mtg30.djvu|30-4]]

The circumference of an ellipse is $$\displaystyle C = 4 a E(e^2)$$    Eq(5)   in page   [[media:Egm6341.s10.mtg30.djvu|30-4]],

Giving the integral:

 * {| style="width:100%" border="0" align="left"

$$\displaystyle I= 4 E(e^2) = 4\int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta $$
 * }

Find Arc length of ellipse using the two Integrals and the integration methods below.

compute $$\displaystyle I_n$$ to the error $$\displaystyle 10^{-10}$$ ,

compute time using tic/toc matlab command and error estimate

1) Composit Trapozoidal rule

2) Romberg Table

3) Clencurt

 * style= |
 * }.
 * }.

Find
verify the result of pb10 of HW5(Team3) and compare with the own code.

Solution
first method $$\displaystyle I( \theta) = \int_0^{2\pi} \sqrt{ {r}^{2}+ {\left( \frac{dr}{d \theta}\right)}^{2}} d\theta$$

When run the Team 3's clencult code, they does'nt porvide answer(while loop continuously run.) the reason is that they use the I_exact value as Quad's solution(6.17660193898874),

but this value is not correct. when they calculate the Quad solution, they didn't designate the Toll error level(e.g. $$\displaystyle 10^{-10}$$). However their clencult code itself is correct,so when I change the I_exact value correctly, they provide correct answer.

Team3

Own work

Romberg Table is correct.

Second method $$\displaystyle I= 4 E(e^2) = 4\int_0^{\pi/2}\sqrt {1-e^2 \sin^2\theta}\ d\theta $$

Team 3 only calculated the quad method.

 Own work

In conclusion, Team 3's work is almost correct except for

1. quad function -- the error order$$\displaystyle (10^{-8}) $$ was bigger than criterion $$\displaystyle 10^{-10} $$ because of not designating error tolerance level in the function

2. seconde method was not calculated except for quad function.

Also, when compare the result bwtween first and second mehtod, the results were same. and in the Romberg table, the second method was faster than first method.(see the upper table)

Matlab Code: method 1-Quad

Matlab Code: method 1-Composit Trap.Rule

subfunction ff5_10

Matlab Code: method 1 - clencult

Matlab Code: method 1 - Romberg Table

subfuntion ff:

Matlab Code: method 2 - Quad

Matlab Code: method 2 - Compsit Trap. rule

subfunction ff

Matlab Code: method 2 - Clencurt

Matlab Code: method 2 - RombergTable

= Problem 5: Identification of basis function $$\displaystyle \overline{N_i}$$ =

Given
Refer Lecture slide 35-2,4 for problem statement

The relationship between $$\displaystyle Z(s)$$ and $$\displaystyle d_i$$(degree of freedom)

 * {| style="width:100%" border="0" align="left"

Z(s)=\sum_{i=0}^{3}c_i s^i = \sum_{i=1}^{4}N_i(s) d_i = \sum_{i=1}^{4} \overline{N_i}(s) \, \overline{d_i} $$ $$ where,
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle \longrightarrow (1)
 * }.
 * }.
 * $$\displaystyle N_i(s)$$ = basis function
 * $$\displaystyle d_i(s)$$ = degree of freedom

$$\displaystyle d_1=Z_i $$ $$\displaystyle \,\,\,\,\, ,d_2=\dot{Z_i} $$ $$\displaystyle \,\,\,\,\, ,d_3=Z_{i+1} $$ $$\displaystyle \,\,\,\,\, ,d_4=\dot{Z}_{i+1} $$

Find
Identify the the basis function $$\displaystyle \left\{ N_i(s) \right\}\,\,\, ,i=1,2,3,4 $$, and plot $$\displaystyle \overline{N_i}(s)$$'s

Solution
From the equation (1) of 35-4(lecture slide)

 * {| style="width:100%" border="0" align="left"

\begin{Bmatrix} c_0 \\ c _1 \\ c _2 \\ c_3 \end{Bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -3 & -2 & 3 & -1 \\ 2 & 1 & -2 & 1 \end{bmatrix} \begin{Bmatrix} Z_i \\ Z_i^' \\ Z_{i+1} \\ Z_{i+1}^' \end{Bmatrix} =\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -3 & -2 & 3 & -1 \\ 2 & 1 & -2 & 1 \end{bmatrix} \begin{Bmatrix} \overline{d}_1 \\ \overline{d}_2 \\ \overline{d}_3 \\ \overline{d}_4 \\ \end{Bmatrix} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

where,

$$\displaystyle \overline{d}_2= h \, d_2$$ $$\displaystyle, \,\,\,\,\, \overline{d}_4= h \, d_4$$ $$\displaystyle, \,\,\,\,\, \overline{d}_1=d_1$$ $$\displaystyle, \,\,\,\,\, \overline{d}_3=d_3$$

Expand the matrix equation.

 * {| style="width:100%" border="0" align="left"

\begin{align} &c_0= \overline{d}_1 \\ &c_1= \overline{d}_2 \\ &c_2= -3\overline{d}_1 -2\overline{d}_2+3\overline{d}_3-\overline{d}_4 \\ &c_3= 2\overline{d}_1 +1\overline{d}_2-2\overline{d}_3+\overline{d}_4 \\ \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

plug in $$\displaystyle c_i$$ to equation (1)

 * {| style="width:100%" border="0" align="left"

\begin{align} Z(s)&=\sum_{i=0}^{3}c_i s^i = c_0 + c_1 s^1 + c_2 s^2 + c_3 s^3 \\ &= \overline{d}_1 + \overline{d}_2 s^1 + (-3\overline{d}_1 -2\overline{d}_2+3\overline{d}_3-\overline{d}_4) s^2 + (2\overline{d}_1 +1\overline{d}_2-2\overline{d}_3+\overline{d}_4) s^3 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

arrange in terms of $$\displaystyle \overline{d}_i $$

 * {| style="width:100%" border="0" align="left"

\begin{align} & = (2s^3-3s^2+1)\overline{d}_1+(s^3-2s^2+s)\overline{d}_2+(-2s^3+3s^2)\overline{d}_3+(s^3-s^2)\overline{d}_4 \\ &=\overline{N}_1(s) \, \overline{d}_1+\overline{N}_2(s) \, \overline{d}_2+\overline{N}_3(s) \, \overline{d}_3+\overline{N}_4(s) \, \overline{d}_4 \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

So,
 * {| style="width:100%" border="0" align="left"

\begin{align} & \overline{N}_1= 2s^3-3s^2+1 \\ & \overline{N}_2= s^3-2s^2+s \Rightarrow N_2= h(s^3-2s^2+s)\\ & \overline{N}_3= -2s^3+3s^2 \\ & \overline{N}_4= s^3-s^2 \Rightarrow N_4= h(s^3-s^2) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Plot of $$\displaystyle N_i $$
 * [[File:basis_function.png]]

= Problem 5: Derivation of Compsit Trap. Rule Error =

Given
Refer Lecture slide 28-3 for problem statement

The Error of Composit Trap. Rule

 * {| style="width:100%" border="0" align="left"

E^1_n= \sum_{k=0}^{n-1} \left[  \int_{x_{k}}^{x_{k+1}}f(x)dx-\frac{h}{2} \left\{ f(x_k)+f(x_{k+1}) \right\}  \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Find
Derive the Compsit Trap.Rule Error equation.
 * {| style="width:100%" border="0" align="left"

E=\sum_{r=1}^{\ell} 2 \, \overline{d}_{2r} \, h^{2r-1}\Bigg[f^{(2r-1)}(b)-f^{(2r-1)}(a)\Bigg]-\frac{h^{2\ell}}{2^{2\ell}} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} \, p_{2\ell}(t_{k}(x)) \, f^{(2\ell)}(x)dx $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Solution
The Error of Composit Trap. Rule

 * {| style="width:100%" border="0" align="left"

E^1_n= \sum_{k=0}^{n-1} \left[  \int_{x_{k}}^{x_{k+1}}f(x)dx-\frac{h}{2} \left\{ f(x_k)+f(x_{k+1}) \right\}  \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Transfer int.$$\displaystyle [x_k, x_{k+1}]$$ to $$\displaystyle [1,+1]$$

where,
 * {| style="width:100%" border="0" align="left"

x(t):=t\frac{h}{2}+\frac{x_k+x_{k+1}}{2}, \, \, \, \, \, \, \, t \in [-1,1]\, \, \,\, \, \,, h:=(b-a)/n $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

g_k(t) = f(x(t)) \,\,\,\,\,\,such \,\,\,that\,\,\,\,\,x \in [x_k, x_{k+1}] $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

 * {| style="width:100%" border="0" align="left"

E^1_n= \frac{h}{2}\sum_{k=0}^{n-1} \left[  \int_{-1}^{+1}g_k(t)dt- \left\{ g_k(-1)+g_k(+1) \right\}  \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

through the successive integration by parts, the below equation can be obtained21-2,326-3

 * {| style="width:100%" border="0" align="left"

\begin{align} E^1_n & = \frac{h}{2} \left[ \sum_{k=0}^{n-1}\Big[p_2(t)g_k^{(1)}(t)+p_4(t)g_k^{(3)}(t_k)+p_6(t)g_k^{(5)}(t)+......+p_{2\ell}(t)g_k^{(2\ell-1)}(t)\Big]_{-1}^{+1}- \sum_{k=0}^{n-1} \left[ \int_{-1}^{1}p_{2\ell}(t)g_k^{(2\ell)}(t) dt \right]\right]\\ &= \frac{h}{2} \left[ \sum_{r=1}^{\ell}\sum_{k=0}^{n-1}\Big[p_{2r}(t)g_k^{(2r-1)}(t)\Big]_{-1}^{+1} - \sum_{k=0}^{n-1} \left[    \int_{-1}^{1}p_{2\ell}(t)g_k^{(2\ell)}(t) dt \right]\right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Extend the first summation term.

 * {| style="width:100%" border="0" align="left"

\begin{align} E^1_n &= \frac{h}{2} \left[ \sum_{r=1}^{\ell}\sum_{k=0}^{n-1}\Big[p_{2r}(+1)g_k^{(2r-1)}(+1)-p_{2r}(-1)g_k^{(2r-1)}(-1)\Big]- \sum_{k=0}^{n-1} \left[   \int_{-1}^{1}p_{2l}(t)g_k^{(2\ell)}(t) dt \right]\right]\\ & because \,\,\,\,\,\,p_{2r}(+1)=p_{2r}(-1),\\ &= \frac{h}{2} \left[ \sum_{r=1}^{\ell}\sum_{k=0}^{n-1} p_{2r}(+1) \Big[g_k^{(2r-1)}(+1)-g_k^{(2r-1)}(-1)\Big]- \sum_{k=0}^{n-1} \left[   \int_{-1}^{1}p_{2l}(t)g_k^{(2\ell)}(t) dt\right]\right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Using below relationship,

 * {| style="width:100%" border="0" align="left"

g_k^{(i)}(t)=\left(\frac{h}{2}\right)^if^{(i)}(x(t)),,\,\,\,\,\ x \in [x_k,x_{k+1}] $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

$$\displaystyle g_k(t)$$can be changed to $$\displaystyle f_(x_k)$$ and below was obtained.
 * {| style="width:100%" border="0" align="left"

\begin{align} &g_k^{(2r-1)}(+1)=\left(\frac{h}{2}\right)^{2r-1}f^{(2r-1)}(x_{k+1}),\\ &g_k^{(2r-1)}(-1)=\left(\frac{h}{2}\right)^{2r-1}f^{(2r-1)}(x_{k}),\\ &g_k^{(2\ell)}(t)=\left(\frac{h}{2}\right)^{2\ell}f^{(2\ell)}(x_{t}) \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

Plug in thess relationships to the equation $$\displaystyle E_n^1$$
 * {| style="width:100%" border="0" align="left"

\begin{align} E^1_n & = \left(\frac{h}{2}\right) \left[ \sum_{r=1}^{\ell} p_{2r}(+1) \left(\frac{h}{2}\right)^{2r-1} \sum_{k=0}^{n-1} \Big[f^{(2r-1)}(x_{k+1})-f^{(2r-1)}(x_k)\Big]- \sum_{k=0}^{n-1} \left[   \int_{x_k}^{x_{k+1}}p_{2\ell}(t_k(x))\left(\frac{h}{2}\right)^{2\ell}f^{(2\ell)}(x) dx \right] \right]\\ &= \sum_{r=1}^{\ell} h^{2r} \frac{p_{2r}(+1)}{2^{2r}} \Big[f^{(2r-1)}(x_{n})-f^{(2r-1)}(x_0)\Big]-  \left(\frac{h}{2}\right)^{2\ell+1} \sum_{k=0}^{n-1} \left[ \int_{x_k}^{x_{k+1}}p_{2\ell}(t_k(x))f^{(2\ell)}(x) dx \right] \end{align} $$
 * $$\displaystyle
 * $$\displaystyle
 * }.
 * }.

by the definition of $$\displaystyle \overline{d}_{2r}$$
 * $$\displaystyle \overline{d}_{2r} = \frac{p_{2r}(+1)}{2^{2r}} $$
 * $$\displaystyle x_n=b, x_0=a $$

 * {| style="width:100%" border="0" align="left"

E^1_n = \sum_{r=1}^{\ell} h^{2r} \overline{d}_{2r} \Big[f^{(2r-1)}(b)-f^{(2r-1)}(a)\Big]- \underbrace{\left(\frac{h}{2}\right)^{2\ell+1}}_{(?)} \sum_{k=0}^{n-1} \left[ \int_{x_k}^{x_{k+1}}p_{2\ell}(t_k(x))f^{(2\ell)}(x) dx \right] $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

This solution is little bit different from the given solution of the lecture.

the coefficient of second summation term that marked $$\displaystyle (?) $$

we got $$\displaystyle \left(\frac{h}{2}\right)^{2\ell+1} $$ instead of $$\displaystyle \left(\frac{h}{2}\right)^{2\ell} $$

= Problem 11: Derivation of the equation of arclength using the cosine law =

Given
Refer Lecture slide for problem statement

Figure 1

 * $$\displaystyle

r(\theta) = \frac{1-e^2}{1-e \, cos(\theta)} $$

where, eccentricity $$\displaystyle e = \sin(\frac{\pi}{12}) $$

Find
Derive the equation of arclength using triangle OAB and cosine law.

 * $$d\ell=d\theta \left[ r^2+ \left( \frac{dr}{d\theta}\right)^2\right]^{\frac{1}{2}}$$

Solution

 * {| style="width:100%" border="0" align="left"

[reference]
 * style="width:20%; padding:10px; border:2px solid #8888aa" |
 * style="width:20%; padding:10px; border:2px solid #8888aa" |

the law of cosine
 * $$c^2 = a^2 + b^2 - 2ab\cos(\gamma)\,$$

 * style= |
 * }.
 * }.

Apply the cosine law to triabgle OAB in the Figure1 (See Figure 1)

 * $$\displaystyle dl^2=\overline{AB}^2 = \overline{OA}^2 + \overline{OB}^2 - 2\overline{OA}\,\,\, \overline{OB}\cos(d\theta)\, $$
 * $$\displaystyle dl = \sqrt{\overline{OA}^2 + \overline{OB}^2 - 2\overline{OA}\,\,\, \overline{OB}\cos(d\theta)}\, $$
 * $$\displaystyle dl = \sqrt{r(\theta)^2 + r(\theta+d\theta)^2 - 2r(\theta)r(\theta+d\theta)\cos(d\theta)}\, $$
 * $$\displaystyle dl = \sqrt{\underbrace{\Big[r(\theta) - r(\theta+d\theta)\Big]^2}_{(a)} + 2r(\theta)r(\theta+d\theta)\underbrace{(1-\cos(d\theta))}_{(b)}}\, $$

Part$$\displaystyle (a) $$ is approximately
 * $$\displaystyle (r(\theta) - r(\theta+d\theta))^2 \simeq (dr)^2 \, $$

For part$$\displaystyle (b) $$

Using the other cosine summation law

 * $$\displaystyle cos(\alpha + \beta)=cos(\alpha)cos(\beta)-sin(\alpha)sin(\beta) \, $$

 * $$\displaystyle cos(d\theta)=cos\left(\frac{d\theta}{2}+\frac{d\theta}{2}\right) =cos\left(\frac{d\theta}{2}\right)cos\left(\frac{d\theta}{2}\right)-sin\left(\frac{d\theta}{2}\right)sin\left(\frac{d\theta}{2}\right)= cos^2\left(\frac{d\theta}{2}\right)-sin^2\left(\frac{d\theta}{2}\right) \, $$

because, $$\displaystyle cos^2\left(\frac{d\theta}{2}\right)= 1-sin^2\left(\frac{d\theta}{2}\right)\, $$

 * $$\displaystyle = \left(1-sin^2\left(\frac{d\theta}{2}\right)\right)-sin^2\left(\frac{d\theta}{2}\right) = 1-2sin^2\left(\frac{d\theta}{2}\right)\ \, $$

In small $$\displaystyle d\theta$$ approximately,
 * $$\displaystyle sin(\frac{d\theta}{2}) \simeq \frac{d\theta}{2} \, $$

using this fact,

 * $$\displaystyle 1-cos(d\theta)= 1-\left(1-2sin^2\left(\frac{d\theta}{2}\right)\right) = 2\left(\frac{d\theta}{2}\right)^2 =\frac{(d\theta)^2}{2} $$

So, Plug in part(a) and Part(b)

 * $$\displaystyle dl = \sqrt{(r(\theta) - r(\theta+d\theta))^2 + 2r(\theta)r(\theta+d\theta)(1-\cos(d\theta))}\, $$
 * $$\displaystyle = \sqrt{(dr)^2 + 2r(\theta)r(\theta+d\theta)(\frac{(d\theta)^2}{2})} = \sqrt{(dr)^2 + r(\theta)r(\theta+d\theta)(d\theta)^2} \, $$
 * $$\displaystyle = d\theta \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r(\theta)r(\theta+d\theta)} \, $$

when $$\displaystyle d\theta$$ is very small, approximately, $$\displaystyle r(\theta)r(\theta+d\theta) \simeq r(\theta)^2 \, $$

 * $$\displaystyle \simeq d\theta \sqrt{\left(\frac{dr}{d\theta}\right)^2 + r(\theta)^2} \, $$

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