Okay. So I've turned on the recording for today's lecture. Welcome everyone. Let's just quickly review where we were, where we paused last week. I will share the notes and let's see. Okay. So those of you who are online, you would have downloaded the lecture notes, the PDF. So please follow along with us. And those are in class. You have the printed version of the notes. So please follow along. So we gave a little introduction last week, just introducing the subject of faith. And we'll quickly review what we did and then we will go forward. So what we said last week is from the introduction that our Christian walk is not a passive life where you just sit and wait for God to drop things on you. It's not a passive thing, but it's an active thing. We have to reach out and take what God has made available to us. We have to reach out. That's why we said in Luke 12-32, Jesus said, It's your Father's good pleasure to give you the kingdom. And yet at the same time in Matthew 11-12 he said, we have to press in. The violent take it by force. So one side, God is very happy to give it to you. On the other side he's saying you have to take it by force, violence. So both are there, both are an important part. And so the taking it by force is where faith comes in to take hold of what is ours in the kingdom. And so we introduced the course and then we began the definition of what faith is. And this is quickly reviewing based on Hebrews 11-1. He said, faith is the substance of things hoped for. It is the evidence of things not seen. So faith is the substance of things hoped for. So first of all we said, therefore we must hope for things. There is nothing wrong in hoping or desiring for things. You must desire to, for example, if God has called you to be a pastor, you must desire to have a church. You must desire to impact the lives of people. You must desire to accomplish great things. You must desire if you're sick, you must desire to be healed. So faith is the substance of things hoped for. You must have that hope or the desire for things. And faith is the substance, it gives substance. We also said it means it's the foundation or it also means it is the title deed. It's the title deed of things you hope for. It's a title deed, it's the proof of ownership. So your faith is the proof of ownership of things you're hoping for. That means it's your proof saying that I have it, it's mine. So somebody says, how do you know you have it? My faith is proof of ownership. And I remember the story that Yonge Cho had written in one of his books. Now, Yonge Cho, this was back in the 1960s, a long time ago. At that time, he was a young man in Korea and there was a Korean war and so the country had gone through a lot of difficulties.  And if you know the story of Yonge Cho, he got used to build the largest church in the history, Christian history, more than 800,000 people in that church. But anyway, this was in the very beginning of his ministry. He shares this little story about when he started in the ministry, he was living in one room, but in that room, he had nothing. There's no chair, no desk, nothing. So he prayed. He said, God, I want a desk, I want a chair, and I want a cycle. I want three things. So he had nothing. He was just in a bare room. He was starting.  And God was teaching him about faith. So he started praying. And then he learned that faith is the title deed. It's the ownership, it's a proof of ownership. That means you have it even though you cannot see it. So when he would tell his friends, come, come and see. I have a desk, chair, and cycle. They'll come to his room. He'll open the door. It's empty. But he said, I have a desk. I have a chair. I have a cycle. But you can't see it. You look all around in the room. There's no desk, no chair, no cycle. But he says, I have a desk. I have a chair. I have a cycle. Where is it? My faith. By faith, I have received it. So by faith, I have already received it from God. But it's not in the room. It's not physically there. You can't see it physically. But he said, come see, come in my room. See, I have a desk. I have a chair. I have a cycle. Nothing there in the room. But he said, I have a desk. I have a chair. I have a cycle. So people thought he's gone. But what he was expressing was his faith. Because faith is the title deed. It's the proof of ownership of things hoped for. He's desiring for that. But you can't see it yet. You can't see it in the natural. But he said, I have a desk. I have a chair. And I have a cycle. I have a spring. I believe in God. By faith, I have received it. I didn't ensure enough shortly after that. I forget what was the time period. Very soon, somebody was leaving the country and they were getting rid of their things. And so they gave him a desk, a chair, and a cycle. And he received exactly what he prayed for. So faith is the proof of ownership. That means you know it's yours even before it actually comes in the natural. So faith is the substance of things hoped for. So he desired, he wanted, he desired, he hoped for a desk, a chair, and a cycle. He had faith in God. He believed he received it even though he couldn't see it. But that was his proof of ownership. And then eventually it came into reality in the natural realm. Okay. So faith is the substance of things hoped for. It is the evidence of things not seen. So even though our natural senses cannot feel it, it's not there. But you have the conviction of the reality. You have the conviction it is true. For example, even when you're believing God for healing, right? You may not feel healed but you believe the Word of God that by the stripes of Jesus you have been healed. Right? You believe you've been healed. No, with me. So far. Okay. All right. So we saw Hebrews 11.1. I'm just quickly reviewing that. Okay. Then our faith connects us to God. So when you want to, you know, connect with God. So Francis, are you using your phone to look at the Bible or? Oh, meaning. Okay. Don't use your phone for other things. Okay. It's not right. So faith connects us to God. Right? So if we want God's spirit, we are in this realm. If we want to connect with God, we have to connect by faith. So we have to connect by faith. Faith is required to please God. So if you want to please God, you've got to have faith. Our faith is in the person of Jesus Christ. We said this, that Jesus is the author and finisher of our faith. So it's not in some mechanical thing or, you know, it's in the person of Jesus Christ. Then we said that faith is based on a relationship. You know, Abraham was called a friend of God. Abraham is a father of faith. And what is, what's he referred to as? He's referred to as the friend of God. So faith is based on a personal relationship with God. And as we grow in our relationship with God, we will also be able to grow and increase in our faith. We also said last week, faith is of the heart. That means with the heart, man believes. So it's not a minded thing. It's a heart thing. So sometimes in your mind, you may have questions. In your mind, there may be doubts. In your mind, there may be, you know, some things that you're not sure of, but you can still have faith in your heart. And that's where the battle many times is.  God is true. God is faithful. God will keep his word. But sometimes you will face questions in your mind. Or things around you will be going against the promise of God. It will be going against, it will be going opposite to the word of God.  You know, it's very opposite. But in your heart, you say, no, God is true. His word is true. I believe it. So faith is of the heart. Even though in the mind, there may be questions. The mind may be stuck, you know, facing and seeing these things that are happening around us. And so there may be questions. It's okay. Right. Don't confuse the questions in your mind with your heart. Your heart can believe even when there are questions in your mind. Okay. So faith is of the heart and we walk by faith, not by sight. And we said that God calls us to live by faith. That means everything you do in life. Do it by faith. Everything.  Then you're going to graduate. Then you're going to start your ministry. You're going to do something. Everything you do, you have to do it by faith. You believe God. God will bless me. God will help me.  You may pray for people or whatever you do, you have to do it by faith. We live by faith. That means everything you are doing in life, you have to do it by faith in God. So our life is a life of faith. And now we've stopped. I think somebody here that our faith is conceived and nurtured by the word of God. Romans 10 17. Faith comes by hearing and hearing by the word of. That's why we must always go to the word of God. And always your faith is based on the word of God, the Scriptures. What did God say? It's based on the promise. So you go back to the promise and read it, meditate in it, be established in the promise of God. Because faith is birthed by the word of God and faith is nurtured and made strong by the word of. So we have to constantly go back, go back and you establish yourself on the word of God, on the promise of God. Because faith is conceived and it's nurtured by the word of. So now faith in the word is faith in God. So when you believe the word, you are believing God. When you believe God's word, you're believing God. So God celebrates that. When he sees you and me, wow, this person really believes my word. He or she is actually believing me. So when you have faith in the word of God, you're actually having faith in. So that's important. We talked about this last week. And faith is like a muscle. It grows as it is exercise. So faith is like a muscle. You have to keep using it, keep using it. And as you keep using your faith, as you keep exercising your faith, it's going to grow. It's going to become stronger. And as your faith is stronger, you can do bigger things, greater things for the kingdom of God. All right, let's start now. So we covered till that last week. So all this was just a quick review. So let's begin now. We're going to look at a few more things about faith. I want us to understand that there are several factors that influence our faith. So we know faith is based on the word. But if we want to exercise faith in God, there are other things that are involved. So for example, if you're doing exercise, you're building your muscle. Very good. But you have to be in good health. If you're sick, then you're laying down, you can't use your muscle. You can't exercise. So your whole body has to be in good health if you're going to be doing something. So it's like that. For faith to work, there are other things involved. What are those other things? In Galatians 5 and verse 6, could somebody read that for us? Galatians 5 verse 6. Go ahead. Just read it out. You can read it from the notes. Galatians are from the screen. Galatians 5 verse 6. Those of you with us online, feel free. If you have any questions, just type it on the chat. And I will come and look at the chat from time to time. And we will be happy to answer your questions. For those of you online, just feel free to type your questions or comments in the chat as we progress. We will look at it from time to time. Galatians 5 verse 6, somebody please read it. So Galatians 5 verse 6. The last part of that verse, it says, what matters is faith working through love. Faith works through love. So what must be there for faith to work? It has to be love. The love of God. You have to be walking in the love of God. So examples. Suppose there's only examples. Suppose Nikhil, right? So Nikhil comes for prayer. Pray for me. Whoever, he may come to you for prayer. Please pray for me. Okay. You want to pray for him. You want to exercise your faith for him or with him, right? But in order to exercise faith with him or for him, what must you have? Love. Because faith works through love. Suppose you just fought with Nikhil. Angry. You're now not talking to him. No, not in good terms. Can you exercise faith? Not possible. Why? Because faith works through love. If you don't have love, if you don't have compassion, it's difficult for faith to work. Understanding. So when we want to pray for somebody, we have to move in the love of God. Because faith works through love. So it's important, right? So faith is based on the word of God. That's important. Faith comes from the word of God, important. But for faith to work, there has to be love. Faith works through love. So I have to guard my heart. Each one of us, we have to guard our heart. And there is no hatred in our heart towards people. And there is no anger. There is no hatred, no bitterness towards people. Because anger, hatred, bitterness towards people can prevent faith from working. So first, suppose, example, suppose I am angry with Nikhil. Before I can pray for him, first I have to get that out. God, I'm sorry. Remove this hatred or anger or whatever towards him. Oh God, take it out of my heart. Fill my heart with love for Nikhil. So now there is love in my heart. Then faith can work. Because faith works through understanding, right? It's very important. And if you look at 2 Peter, 2 Peter chapter 1 verses 5 through 8, Peter in that passage is telling us to faith, to the faith we have in God, we have to add many other things. So I want you to count how many other things to add to faith. Just count in those verses. He says to your faith, add, add to your faith. Let's count it, virtue, to virtue, knowledge, to knowledge, self-control, to self-control, perseverance, to perseverance, godliness, to godliness, brotherly kindness, and to brotherly kindness, love. So how many? Seven. So to faith, you keep adding these things. He mentioned seven things. That means faith needs this kind of an environment. It needs to be surrounded by these things, the seven other ingredients for it to thrive, for it to grow and be nurtured and become strong. Right? So add to your faith seven things he mentioned in a virtue, knowledge, self-control, endurance, godliness, brotherly kindness, love. Right? So why am I emphasizing this? Sometimes when we talk about faith and we are focusing in this whole course, we are focusing on this one topic of faith. But remember, other things are important around it. Right? For faiths to work, there has to be love. For faiths to be in a good condition, there has to be these other things like virtue and knowledge and brotherly kindness and self-control and endurance, all those things need to surround the faith that we have in our hearts for faiths to be really effective. Okay? We are going to spend almost all our time talking about faith. But don't forget, other things are also important around it. Okay? So you can keep this list in mind that faith needs these other things to also be built up and encouraged. And in 1 Timothy chapter 1 and verse 19. What was that? Somebody's phone? Okay. Sorry, there's some interruption here. Okay. In 1 Timothy chapter 1, verse 19, Paul the apostle, he mentions this, that we need faith. We need a good conscience for faith. I mean, that conscience must be clear for faith to work. So that's another aspect that we must keep in mind. Right? So let me just pause and see if anybody on the online class has any questions. All right. Okay. So I see a question here from Nina. When we say heart, is this used interchangeably with man's spirit? That's right. So Nina, the answer to your question is yes. Right? So Nina's question is, does heart mean spirit? Okay. So in the New Testament, it's almost synonymous. That means heart and spirit are used in a synonymous way. They're interchangeable. But in the Old Testament, the heart is often used inclusive of spirit and soul. If you study the way the word heart is used in the Old Testament, it kind of inclusive spirit and soul. But in the New Testament, for some reason, you will find a demarcation between spirit and soul. And so the heart is used in a very distinct manner. So to answer your question, in the New Testament, heart and spirit can be used interchangeably. In the Old Testament, you've got to keep in mind that sometimes the word heart, refers to the inner person, includes the spirit and soul. Okay. All right. There's another question here from Krishna. Is someone who hurt us emotionally, but we are still praying for them? Does it affect our faith? Krisha. Krisha. Yeah. Sorry, Krisha. I said Krishna. Thank you for correcting me. So Krisha has a question. If someone has hurt us emotionally and we are still praying for them, does it affect our faith? You see, as long as we are able to love them, so people hurt us, of course, it causes us pain. We may feel upset and all of that. But if we forgive, if we let go, we are able to walk in the love of God and then we pray from that place of love and forgiveness, our faith would be effective. But if somebody has hurt us and we are angry, we are hateful, we are resentful, bitter towards them, then that negative emotion of hatred, unforgiveness and good towards them is going to hinder our faith. Okay. So it's so important that we first forgive. We first release the love of God and then begin to extend our faith towards them. Okay. And a classic passage is in Mark 11, 22, 23, 24 and 25, Mark 11, 22 to 25, because Jesus says, have faith in God. You can move the mountain verse 23. You can pray and receive from God. And right after that, He tells us to forgive. So it's all connected and it shows us the importance of forgiving people. I hope that answers your question, Krisha. Thank you. Good. Thank you for those questions and any other questions. All right. Let's get back. Any questions from those of you in class? Any questions? Okay. All right. Let's get back to the notes now. I'll go ahead and share my screen. So let's move forward. Another important aspect about faith. Faith causes the power of God and His word to be released. This is very important. Faith causes the power of God and His word to be released. In 2 Thessalonians chapter 1 verse 11, notice the Apostle Paul is praying for these believers. He says, I'll be praying for you that our God would count you worthy of this calling and fulfill all the good pleasure of His goodness and the work of faith with power. He's saying we are praying for you. So just look at the last part of that verse. That God would fulfill the work of faith with power. To fulfill means to complete. That God would complete your work of faith.  You're acting your faith. You're working your faith. You're putting your faith to work. What will happen? God will fulfill your work of faith with... But if you don't work your faith, His power is not going to be released. So a good example which I've been mentioned last time is that faith can be imagined like a switch. So there's power coming in those wires from the electricity board. It comes in all the way. And there's a switch. But if the switch is off, the power is stopped there. It's not flowing through. But when you put it on, power flows. So just as an example, faith is like a spiritual switch. So when you work your faith, when you act your faith, when you do something with your faith, then God fulfills or He completes our work of faith with power. So His power goes into operation when we work our faith. Not when we sleep on our faith. We work. We have to exercise our faith. Then His power goes into operation. So faith causes the power of God and His word to be released. There's some more scriptures on this. Look chapter 1 verse 45, when Mary, the angel has come to Mary and said, Mary, you're going to have a child by the power of the Holy Spirit. So it's a really shocking word. And then in that context, this is what is spoken to Mary. Elizabeth speaks this by the Spirit. She says, Blessed is she who believed. For there will be a fulfillment of those things which were told her from the Lord.  When you believe that, then you will see a fulfillment. So God has spoken. We believe it. Then there is a fulfillment of that word. So that's what Elizabeth is saying. Blessed are you who believed because there will be a fulfillment of those things which were spoken to you by the Lord. So faith is what causes that word for God's power to be released for that word to be fulfilled. One more scripture on that, Hebrews chapter 4 and verse 2. It says, For indeed the gospel was preached to us as well as to them referring to the Old Testament people, people of Israel. But the word which they heard did not profit them, not being mixed with faith in those who were dead. So think about this. The word of God is full of the power of God. Correct. His word is alive and powerful. So God's word is powerful. There's nothing wrong with it. God's word is not empty. God is watching over his word to fulfill it. But it's telling us in Hebrews 4 verse 2 that the word did not benefit them. It didn't do anything. So can you imagine God is sending his word and his word is not accomplishing anything in them. He's referring to the Old Testament people, the people of Israel at that time. It didn't do them any good. The word was given to them. It didn't profit them. Why? It was not because there was a problem in the word. The word was pure. The word was the word of God. It was holy.  What went wrong? It says it was not mixed with faith. That's the problem. The word came. They didn't mix it with faith. They didn't mix it with faith. So what happened? It did not benefit. They heard the word. God gave them the promise. But still it didn't benefit. It didn't benefit. So if you want to put it in a positive way, when we hear the word and we mix faith with it, we have faith in the word, then faith in the word causes the power of God to flow into our lives. So you hear the word of God and you believe that word. God, I believe that what you have said, I believe it. Then the power of God will flow into your life. So example, example. Suppose as a believer, I believe in Jesus. I believe God. But suppose right now in my life, example, things are very difficult. I'm struggling in my life. Financially, I may be very difficult, but things are not going very well in life. Example. Suppose that's my condition. Then I open my Bible and in the Bible, I see, I just open my Bible. I say, okay, please let me read one Psalm today. I'll read the first Psalm. Okay, Psalm 1. Blessed is the man who walks not in the council of the ungodly. He doesn't stand in the way of sinners. He doesn't sit in the seat of the scornful. But his delight is in the law of the Lord. And in his law, he meditates day and night. And he shall be like a tree planted by the rivers of water. He will bring forth his fruit in its season. His leaf will not wither and whatever he does will prosper. Hallelujah. Jump from my place. I start dancing. God, you said in your words, I will be like a tree planted by the rivers of water. I will bring forth my fruit in its season. My leaf will not wither, whatever I do will prosper. Then I start dancing, shouting, Hallelujah. Right now, things are bad in my life. Right now, things are very difficult. But God has given his word. And he says, if I fear the Lord, if I delight in this word, if I don't do the wrong things, I will be like this. I will be like this. So what happens? I mix faith with that word. And I begin to say, I am like a tree planted by the rivers of water. I bring forth my fruit in its season. My leaf does not wither, whatever I do will prosper. What is that? Around you, things are very bad. Right now, things are very difficult. But it doesn't matter. Because God has given his word. And when you mix faith with that word, what will happen? That word will start working in your life. That word will start releasing God's cover into your situations, into your circumstances. And things in your life will begin to change. It will begin to change. Things will happen. He said, how is it happening? It's because the power of God is working in your life situation. And the word of God will transform your life situation. And you will become like that tree. You will be fruitful. You will be evergreen. And whatever you do will start prospering. Amen? That's how it will happen. Or imagine somebody reads the Bible. Okay, today I have to do a devotion. Psalm 1. Read Psalm 1. Halfway you sleep through the Psalm. You reach the last verse. Close the Bible. You go to... You read the same Psalm. But it didn't profit. It didn't benefit. You read the same Psalm. Maybe even highlighted yellow color, blue color, red color. I don't know. But you didn't mix faith with that word. You read the same Psalm. One person read the Psalm believed that Hallelujah, it is done. And started rejoicing. I said, God, I believe that word. And that one word from God can transform your life if you mix faith with it. Another person read the same Psalm. Colored it. Different colors. Close the Bible. So I did my devotion. Yeah, you did your devotion, but you didn't believe the word. You didn't mix faith with the word. So what happened? The word that you read or the word that you heard didn't profit, didn't benefit. No, didn't benefit. Why? The problem was not with the word. It is from we didn't mix faith with the word. Are you understanding? So in every... This is so important. This is why faith is important. We must mix faith with the word of God. With what we share. And God's word will produce. It will work in our lives. Now I'll just tell you to give you one story and then we'll go for a break. So I was doing my engineering. I was studying. This is for my masters. So I'd gone to the US to do my masters and I had made a decision in my heart that I will not ask my parents for money after going there. I'm going to study on my own. I'm not going to ask my parents for money. And this was long time ago. So this was the beginning of 1992. I was in the university there. And that semester... The semester fees was I think around $6,000. I forget the exact amount. Something like that. $6,000 for that semester. One semester every semester. So for that semester fees was $6,000. I have to pay. I was also working. As a student you're allowed to work on campus. I was also working. But that was only a stipend. Meaning I would only get $800 a month which was enough for food and rent. I didn't have any more money to pay the fees. But I said, I will have faith in God. I will have faith in God. So I started meditating in all these scriptures. Psalm 23 verse 1. The Lord is my shepherd. I shall not be in war. My God will supply all of your need according to his riches in glory. This is the Bible. This is the word of God. It's not true. It is the word of God. So God you said you will supply all my need according to your riches in glory. So like this I took seven scriptures. Also Matthew 17 20 said, if you have faith in your heart like a mustard seed, you speak to the mountain. Tell it to move. It will move. Nothing will be impossible for you. I said, God, I'm speaking to this mountain. My tuition bill. I declare. So every day I will get up. Semisters started. Okay. They give you time to pay the fees. So semester started. I'm attending the classes, but no money to pay. I don't have money yet to pay the fees. Right. And I'm every day I'm declaring the word of God. And I speak to my tuition bill, tuition bill in the name of Jesus. You are paid in full.  Tell it to move. It will move. So I said, every, I command this debt, this mountain of debt to be canceled. My tuition bill is paid in full in the name of Jesus. So speaking, speaking pretty. Now, semesters going on. I remember in class one day, a professor, he takes up the list. He says, I just, your name is here. You still haven't paid this tuition fees in class. I said, yes, I'm working on it. Which was true. I'm working by faith. I didn't tell him I'm working by faith. I just said, yes, I'm working on it. Okay. The time is running out. You have to pay the fees. And if you don't pay the fees, then by the end of the semester, then whatever you've done in the semester is not counted. You know, you would have written all the exams. You've done all the assignments. But if you don't pay the fees, it's not counted. It's not regarded as though you've done all the courses. You know, it's a waste. So time was, you know, time was going. I was, I'd crossed, I think we'd crossed the middle of the semester.  What is going on? Where is the money going to come from? They will send you back to India. You won't have a degree. And it'll be very shameful.  True. But in my heart I said, no, God said in his word, my God will supply. So the louder these thoughts got in my mind, the louder and stronger my faith becomes. God she will provide. I don't know how that God will provide. I remember one day I was doing work for one of the professors. So I went to the, I was working, doing my work in the lab. And suddenly the professor came. He said, I have received a fax. I got a fax. That means, you know, electronic transmission. Got a fax. He got a fax from the head of one department, another department for whom we were doing research. And he said, that person sent me this fax. On the fax, he mentioned that we have about $6,500 extra for this financial year or this academic year, sorry. And he's asking my professor, the professor for whom I was doing research, if he would like to use that money anyway. So the professor came to me, showed me the fax. He said, I'm going to tell him, we will use this money to pay your fees. I was, wow. You know. And then they gave me the check to pay my fees. I was so proud. But this was, for me, it was like God fulfilled his work. He fulfilled his work. How he did it, that was up to him. I didn't even know that, you know, all these things would happen. I didn't know. My part was to mix faith with the word, to believe the word of God and to stand firm in faith. God's part is fulfill the words. So when you have faith, he will fulfill your work of faith with power. Understand it. So that's how you and I must exercise our faith in God. And not only did I get money for that semester, he also said from now on, we will cover all your future studies covered. So not only that, everything was taken care of. So just amazing how God brought all of that together. All right. So let me just check quickly any questions here on the chat before we go for a break. All right. Okay. All right. So let's take a quick 10-minute break. We'll go for a break. We'll come back in 10 minutes. See you online. I'll see all of you in 10 minutes. Just a quick break and we'll be back. Thank you. God bless.


Let's consider what happens when we row-reduce a square matrix A. Remember, we can perform elementary row operations by multiplying by an elementary matrix. And the elementary matrix corresponding to a row operation is the matrix produced when the row operation is applied to the identity matrix. For example, let's suppose E is the elementary matrix corresponding to multiplying the third row of a matrix by negative 2 and adding it to the fifth row find the determinant of E. Now, notice that we don't know the size of E except that it has at least five rows and columns. But since the third row is added to the fifth, only the fifth row changes. So we know the first four rows, and if we multiply the third row by negative 2 and add it to the fifth, we get the new fifth row. And if there are more rows, all other rows of the identity matrix are unchanged. Since this is a lower triangular matrix, the determinant is the product of the entries along the main diagonal. But since this was an identity matrix, those entries are all 1s, and so the determinant of this elementary matrix is going to be 1 as well. By a similar analysis, we find that the elementary matrix corresponding to switching two rows has determinant negative 1, the elementary matrix corresponding to multiplying one row by a constant c has determinant c. And we just determined that the elementary matrix corresponding to multiplying one row by a constant and adding it to another row has determinant 1. We can show that if E is an elementary matrix, then the determinant of Ea is the determinant of E times the determinant of A. But you should show it for row switching and for multiplying a row by a constant. Now this is also true if E corresponds to multiplying one row by c and adding it to another. This is a little harder to prove, and so we'll consider an example which we can generalize. And by we, I mean you. So we have our matrix and let E be the elementary matrix corresponding to multiplying the first row by 3 and adding it to the second row, let's find the determinant of Ea using only the defining properties of the determinant. So we'll go ahead and compute Ea, which will be. Now remember the determinant is linear in the entries of a row or column and so our determinant of Ea can be expressed as the sum of two determinants, but which ones? And the key here is only change the row that's changing. And since the second row was the one that changed, we'll keep the first and third rows the same. Since we'd like to relate the determinant back to the determinant of A, we'll make our first matrix A itself, which means the second row of the other matrix must be. So let's consider this second determinant. Now just that the second row of the second determinant is three times the first row and if you multiply one row of a matrix by a constant, you can factor out that constant. And so the determinant of the second matrix is three times the determinant with the three factored out. And since this matrix has two rows that are the same, each determinant is zero. So over on the left we have the determinant of Ea, over on the right we have the determinant of A plus zero, or just the determinant of A. Or since the determinant of E is one, we again have the determinant of Ea equal to the product of the determinants. So putting our results together, if E is an elementary matrix corresponding to a row operation on a square matrix A, then the determinant of Ea is the determinant of E times the determinant of A. Now Lather rinsed repeat to get the following. Let E1, E2, and so on be elementary matrices of the appropriate size where the product with A is equal to R. Then the product of the determinants is equal to the determinant of R. And this is an important result, and we'll see where it takes us next.


Dylan Schumacher, Citadel Defense. So I made a video a couple of weeks ago regarding secondary crime scenes. And in that video, you can go back and watch part one. I said, never, never, never, never go with someone to a secondary crime scene because nothing good ever happens there, which is true. And I thought a little bit more about that. And I think I wanted to explain why a little bit more. So there are two types of crime as we break them down, right? There is for-profit crime, right? Give me your wallet or your TV or your car or whatever. Like, I'm committing a crime as a means to an end. The thing I want is something else. That's not you, right? I want your stuff. It's a for-profit crime, right? Again, your car, your wallet, your purse, whatever.  And nobody needs to get hurt here, right? We've all seen that. So that's the for-profit crime. And then there's the second kind of crime, which is thankfully more rare, but it still exists, where it's you are the crime, right? That's the rape. That's the torture murder. That's the child molestation. Those are the abductions. Like, that's the really sick, dark, twisted stuff where you are the crime. Now, the problem is a lot of people can confuse the two. And usually, of course, the confusion becomes, look, just give them what they want and then they'll leave you alone, right? We assume that all crime is for-profit crime. Now, granted, I will say, profit crime is the majority of crime, right? Again, thankfully, the really sick, dark stuff is on the rarer side. However, it still exists, right? And that's why we say, never go to someone with a secondary crime scene, because that's a tip-off that it's not the for-profit crime. If they just wanted your stuff, they would have taken that already and left. But if they want you to go somewhere with them, that's the same thing that, right? If someone takes your purse and takes it with them, they've accomplished their goal. If they take you and go with them, they've accomplished their goal, right? That's what they want. That's where you become the crime. So I think that's an important distinction. And it's good not confuse the two, because if you're dealing with the one who's the second kind, where you're the crime kind of person, right? And you're like, here, just take my wallet, but that's not what they want. And they might take your compliance, meaning that you're gonna be a further easy target and this is gonna be a good time for them, right? So is there an easy way to tell right away when someone approaches you and starts committing a crime? Not necessarily, no. And for that reason, right? That's why we carry guns to defend our lives. Because when someone puts our life in danger, whether it's a for-profit crime or you are the crime, your life is in danger, period. Ability, jeopardy, opportunity, right? So I just want to make that distinction and have that clear in your mind. There are two kinds of crime up there. There's the for-profit and then there's the stuff where you're the crime. And the stuff where you're the crime is the really sick and dangerous stuff. And that is why we never, never, never, under any circumstances ever, go with someone to a secondary crime scene. Do brave deeds and endure.


Cat flu is highly contagious and can spread from one feline friend to another via eye, nasal or mouth discharge. Contaminated food or water bowls can also be a source of the spread of disease. Some infected cats don't show any clinical signs, and are known as carrier cats. A carrier is only infectious to other cats when they are shedding the virus, which may be continuous or intermittent. Prevention can be achieved in most households by vaccination. As there are many different strains of the virus, vaccination may not prevent your cat from becoming infected, but will markedly reduce the severity of the disease.


So we are going to look at property relations and in property relations we will consider relations between properties of a simple compressible system that is a fluid system. So one two way mode of work and that is expansion PDV. We will generally assume a unit mass of the system. So we will be working in terms of specific properties small s, small u, small v, p anyway it is small and t will always be capital. The tools which we are going to use is and if a few of you are uncomfortable with tools we will spend some time on it. The calculus of exact differentials we will come to it when we need to know what it is about. Calculus of partial derivatives related to it and we are going to look at energy functions. Everything is a play on energy functions. By energy functions we mean the energy itself and since we are going to consider actually here I should have said simple compressible fluid system at rest. So by default we are going to assume that delta E or DE will be replaced by DU. Although we know one energy that is the thermal internal energy U which is defined by first law. For convenience we have also defined other energy functions enthalpy which is U plus PV. We call it an energy function because it is a definition using the definition of the internal energy U and there are two more functions which we will define for convenience and all this set of four functions U, H and the two more to be defined will be known collectively as energy functions. And why do we define them because it is helpful to define them just because we found that U plus PV is a common sub expression which we come across. So we give it a short form enthalpy so we will do with two more energy functions. So the first law defines U, second law defines S the entropy and you have already seen for a simple compressible system the basic property relation which relates the two. Remember that this relation TDS is DU plus PV is a that is the basic property relation. It includes first law it includes the definition of entropy. We will also use the first law. Now here the first law is DQ equals DU plus DW where DW includes all modes of work. Whereas in the property relation the left hand side becomes TDS because we are considering a small elemental reversible process linking two neighboring properties and naturally because it is a reversible process DW has to be nothing but PV that is the difference. We will also be using the second law which dictates that TDS is greater than or equal to DQ or TDS is greater than or equal to DU plus DW. Now here if you have not realized already what you can do is the following. Compare this equation property relation with this equation. TDS has to be greater than or equal to DQ. On the right hand side DU is the same what can you say about PVV and DW? If the left hand side TDS is higher than DQ, PVV has to be higher than DW. But DW is PVV plus some other component say stirrer work or electrical work which for a fluid can only be one way work. So that work component has to be what? Always has to be less than 0 and that means for a fluid system work other than PVV work for example stirrer work which will all be one way work modes will always be such that you can do work on the system. System cannot ever do the work on some mode by which one way work is done. So that explains our observation that if you put a stirrer in a fluid, fluid is initially at in a state of equilibrium whatever you do the fluid will not be able to stir the stirrer. The stirrer will be able to stir the fluid. Similarly the fluid will have some electrical resistivity. So you can put two electrodes and pass a current and you can provide electrical work input to that fluid. But you put two resistors and say push a current through it, well it cannot because you know it does not have the capability of getting charged or discharged. It is not an electrolyte. So now look at we will now start playing with our energy functions. The internal energy u related to dq and dw by first law and if I expand dw it will become du plus pdv plus dw other that is what I have said. We have done this yesterday and if I consider a constant volume process for which pdv is 0 then dq at constant volume is du plus dw other. This also we know nothing special here. So let us go away quickly. Similarly in a way enthalpy can be treated h is defined as u plus pv. So expand it dh is du plus pdv plus vdp. Now du plus pdv can be replaced by dq. So we get dq which is du plus pdv plus dw other. This is the same relation which is here. Can now be written down as dh minus vdp plus dw other and if I consider a constant pressure process which makes dp equals 0 then dq at constant p is dh plus dw other. Fair enough? Nothing no surprises here. This is all known. That is because we are using only first law. Yes sir? No significance. It is an algebraic term. We will discuss that offline that earlier when we used to have reciprocating compressors. If you consider one complete stroke or one complete cycle because of the associated flow work involved the work per cycle can be shown to be equivalent to integral of minus vdp. But that is under very very special conditions. But now we define two more energy functions. The first one of them is known as the Helmholtz function. In mechanical engineering we do not routinely use them but when it comes to physical chemistry these are routinely used. These two functions are named after two great contributors to thermodynamics Helmholtz function and Gibbs. The Helmholtz function usual symbol is a but traditionally the symbol used to be f. So you may still find and also the Helmholtz function and Helmholtz Gibbs function. Other names are Helmholtz energy or Gibbs energy and sometimes Helmholtz free energy and Gibbs free energy. It is defined as u minus ts. So s enters in the definition unlike in enthalpy where it is u plus pv and because for u it is defined by first law and pv does not have entropy. So we do not have to use second law when we played with du and dh as on the previous slide. So if you differentiate it you will get da equals du minus ts minus s dt replace du by dq minus dw transpose terms you will get dq minus ts on the right hand side which by the second law is less than or equal to 0. Remember ts is greater than or equal to dq. So dq minus ts would be less than or equal to 0 transpose terms and you will get dw is less than or equal to da minus s dt. I am neglecting this part I am saying this less than or equal to 0 means da plus dw minus s dt is less than or equal to 0. So dw is minus less than or equal to minus da minus s dt and work done in an isothermal process would be less than or equal to minus dA t. And why do we get that inequality here because we have to invoke the second law and what does this mean? This means that A the Helmholtz function is similar to a potential the decrease in which represents the maximum work that can be obtained in an isothermal process. Compare this with water at a height it has a potential for some body m of mass m in gravitational field G is at a height with respect to some datum of h1. We reduce its height to h2. So h1 minus h2 is the decrease in potential energy that is the maximum work we can extract out of it. If there is no friction while it is decreasing that will be the work which will be extracted out of it. In a similar fashion A can now be considered to be a potential. So we can even call it Helmholtz potential. The decrease in which represents the maximum work that can be obtained in an isothermal process. Now the last of the energy functions which we are going to look at is the Gibbs function. In fact this is more important than or more popular and more celebrated than the Helmholtz function. It is defined as u plus pv minus ts everything is put into it. U pv ts all properties are included in it. Of course using the definition of enthalpy it can also be written as h minus ts. Differentiate du plus pdv plus vdp minus tds minus sdt. Use first law so you will get dq minus dw other plus vdp minus tds minus sdt transpose term to get dq minus tds on the one side which by second law is less than or equal to 0. And this means dw other other than pdv work will be less than or equal to minus dg plus vdp minus sdt. And that tells us that if we have a process which takes place at constant pressure and constant temperature then dw other during that process has to be less than or equal to minus dg at pt. So what is the meaning of that? The meaning of that is the Gibbs function can be considered as a potential. The decrease in which represents the maximum work other than expansion that can be obtained in an isobaric or isothermal process. And this is important because many of our chemical reactions start point is one atmosphere 30 degree C. We want end point also to be one atmosphere 30 degree C. So during this reaction what is the maximum energy that we can extract as work? So for fuel cells, electrolytic cells it is the Gibbs function which is of importance. And again the inequality here turns out to be the inequality due to the second law of thermodynamics. Now chemists and chemical engineers would use this idea of Gibbs function very heavily always they cannot live without Gibbs function. But unless we go into combustion and kinetics of combustion we do not really have to worry about Gibbs function. We will use these four functions as a handle to obtain some very important property relations. And for that we have to use some characteristics or some properties of exact differentials. So we will now look at the properties of these four functions but we will be using the specific property version. Whatever I showed you in all these slides was capital properties. So now we will divide everything by the mass of our system. Everything will be written in terms of specific properties. We will use the property relation and also the following formula. And this is one major characteristic which we should remember that if z is some function of x, y and dz can be written down as m dx plus n dy and it is an exact differential. And if z is a proper function what they say analytical function of x and y dz will be an exact differential. We should remember that property is a function of state. So for us it is an analytic function of other properties. So all properties differential of properties will be exact differential that is important for us. Then m and n can be represented in terms of partial derivatives of z with respect to x and y. Now y am I writing this y subscript here and x subscript here. The reason for that is simple. The reason is when it comes to mathematics when you write some f of x y or z of x y partial of with respect to x y is assumed to be constant. There is nothing special about it. But in thermodynamics we have a choice of selecting any two variables as our independent variables. Suppose you select z to be entropy for a fluid system. x and y can be p and v can be p and t can be p and u can be h and t any combination. Particularly if I select x to be t y can be volume pressure enthalpy any of the other functions that I have defined except entropy because entropy is the z part. So I can have variation of entropy with pressure at constant volume variation of entropy with temperature at constant volume at constant pressure. So I have to be specific about what is the pair of properties that I am using. So when I write partial of z with respect to x at constant y that means I am considering z as a function of x and y and then differentiating it with respect to x keeping y constant. So if I write partial of s with respect to t at constant v that means I am considering t and v as independent variables considering the property s as a function of t and v and then differentiating s with respect to t by keeping v constant. Similarly I can have a different derivative partial of s with respect to t at constant p. Here the whole basis is different. I am considering t and p as independent variables to make that clear with subscripts like x and y will always be used in thermodynamics and also the cross derivatives are independent of the order. So the second derivative of x first with respect to x keeping y constant and then with respect to y keeping x constant is equal to the second derivative obtained in the reversed order and because of that the partial derivative of x with respect to y sorry partial derivative of m with respect to x with respect to y with constant x and partial derivative of n with respect to x at constant y because both of them would be second derivative of x second cross partial derivative of x order of differentiation is an important. So these two or three relations are of utmost importance to us. Later on we will look at the properties of Jacobians. We will come to that when we need to use them. So now we really go through our basic exercises in the next four slides. We look at the internal energy u. We know that it is tds minus pdu property relation. Tds is du plus pdu. Now notice on the left hand side we have du on the right hand side we have ds and dv. So we consider that for this equation that equation tells us that for u s and v are very convenient independent variables. So we consider u as a function of s and v and because of that we can straight away write notice the factor here of ds is t. So t must be partial derivative of u with respect to s at constant v. Similarly the coefficient of dv is minus p. So minus p must be partial derivative of u with respect to v at constant s and many people consider these t and p as important. These two relations are considered very important and they are considered to be the thermodynamic definition of t and thermodynamic definition of p. Because t here is related to thermodynamic variables u, s and v. Remember my very first lecture that any system will have at least one property and that is volume because the boundaries are defined. u comes out of first law, s comes out of second law. So this is a quantitative definition of t with respect to u, s and v. And pressure we have defined using fluid mechanics but here we have a thermodynamic definition of pressure which has nothing to do with forces or anything. It is just the partial derivative of internal energy with respect to volume at constant entropy with a negative sign. So these two relations are often considered as thermodynamic definition or thermodynamic basis for temperature, thermodynamic basis for pressure. But what is more important is this is the m and n relation. But now let us look at the cross relation. This means the partial of temperature with respect to volume at constant entropy will be partial of pressure with respect to entropy at constant volume of course with a negative sign. And this relation is an esoteric relation. We never expected it. It relates variation of temperature with volume at constant entropy to variation of pressure with entropy at constant volume. This such we are going to derive three more such relations and these three relations plus this are going to be called Maxwell's relations or Maxwell's equations. This is one such equation based on the internal energy u. Now this is algebra but after going through the four slides of algebra we will come to a basis and we will understand what these four Maxwell's relations really represent. So let us continue. We consider enthalpy h and expanding dh as a differential and then using the property relation we obtain dh is tds plus vdp. All that I have done is du has been expanded as tds minus pdv. Now that means h is related to ds and dp. So we consider h as a function of s and p. And hence t must be partial of h with respect to s at constant p and v must be partial of h with respect to p at constant s. And the cross derivative gives us this second Maxwell's relation. Again notice temperature pressure entropy volume entropy pressure. Some funny combination between thermodynamic variables is dictated by our property relations algebra and calculus. We get two important relations when we work with Helmholtz function and Gibbs function. In this slide as well as in the previous slide you will notice that entropy occurs on either side here as well as here. Similarly on this slide entropy occurs on the left hand side as well as the right hand side. But when we come to Helmholtz function and we differentiate it and use the property relation notice here that du has been expanded as tds minus pdv. You get da equals minus sdt minus pdv. So natural independent variables are t and v. We consider a as a function of t and v. The first partial derivatives give us s and p as partial derivatives of a. And the Maxwell's relation which comes out of cross derivatives is partial of s with respect to v at constant t. Notice on the right hand side we have only pvt no entropy no enthalpy no internal energy. So the third Maxwell's equation this equation which comes out of a is important because it relates the variation of entropy with volume at constant temperature purely to pvt data that is the equation of state and to nothing else. So although entropy is an energy function defined in a wake dq by t reversible. Its variation with volume at constant temperature is related only to pvt to our equation of state and to nothing else. It is not related to cp, cv, nothing because we know pvt data has nothing to do with the heat interaction. It is just an equation of state data. Similarly the fourth Helmholtz function based on the Gibbs relation function which is h minus ts expand dg here I have used dh is tds plus vdp which we have already derived. So dg comes as minus sdt plus vdp. So natural independent variables would be temperature and pressure. So the first derivative give us entropy and volume as partial derivatives of Gibbs function but the Maxwell's relation which comes out of it gives us the variation of entropy with pressure at constant temperature only in terms of pvt information. So the last two Gibbs functions are important. They are so important that many people tend to simply many students would be doing is simply you know learn this by heart or have a small cheat somewhere in their pocket listing this down. So called cocksheet or they have some mnemonic or some shloka which puts all these things in order. These things are important particularly the last two are very important because they relate entropy, variation of entropy with volume and pressure respectively to purely pvt data. pvt data is the simplest data to measure. No cp, cv is required. So it is important to be able to use this and to use this properly we must be able to remember this and we will be able to remember this if we are able to derive them very quickly. And for that we will now look at the thermodynamic basis for these. Before that these relations as well as their reciprocals. For example you take this relation partial of s with respect to v at constant t is partial of p with respect to t at constant v. You can go through the calculus and this becomes equivalent to saying partial of v with respect to s at constant t equals partial of t with respect to p at constant v. We have analytic functions so there is absolutely no harm in saying that. So these relations and their reciprocals are known as Maxwell's relations and they are very useful property relations. And as I have already said the third and fourth ones relate entropy variation to purely pvt data. And they help us reduce the requirement of cp or cv data for mapping the state space. Now how do we remember them? We come to an absolutely basic stuff in thermodynamics. Let me go to this. Let me ask you the following question. Let us say that I have a fluid system and I consider a process some process and I say that I consider a process from 1 to 2 which is a quasi-static process. I have shown it in a continuous line on the pv diagram so this is a quasi-static. What does the area under the curve represent? This represents expansion work done. Now I consider a TS diagram and consider a process maybe the same process quasi-static maybe it gets represented like this. It also is a quasi-static process. What does the area under it represent? Does it? See it represents integral Tds. Just the way this represents integral pdv which comfortably we say uneditingly is work of expansion. But this represents integral Tds which we will be able to evaluate because it is a defined curve. dq is what? dq is see Tds is greater than or equal to dq. So dq is less than or equal to Tds. So the q will have to be less than or equal to this area. When will it represent the area? If it is a reversible process. So a TS equals dq only if process is reversible. So remember the difference. This area is q only for a reversible process. It is integral Tds alright. Integral Tds will be dq only for a reversible process. Whereas integral pdv quasi-static process you are able to evaluate it so it is W expansion. No doubt. So get this idea and now let us go back to our, I considered a process. Now instead of a process I consider a cycle executed by a fluid system and let us say I am considering a reversible cycle. So it is executed in an absolutely reversible way. Otherwise there is no restriction on the processes which could be there in the cycle. It could be a small cycle, dp dv type or it could be a big cycle taking it anywhere it feels like. We represent it on a pv diagram. We also represent it on a TS diagram and we will see something like this. We will see a pv representation and we will see a TS representation. It is a reversible cycle. So the closed area of the pv diagram of the loop in the pv diagram will represent work done per cycle or in that cycle. Will it be total work or will it be pdv work? But I have said it is a reversible cycle. It is a fluid system. So the only work done will be pdv work. So this would be the work done in the cycle which also is the pdv expansion work because it is a reversible cycle no other mode of work is admissible. It is a reversible cycle. So the area on the TS diagram is heat absorbed during the cycle, net heat absorbed during the cycle. Are the two areas equal? Yes they are. What does it mean? We said they are equal because cyclic integral of pdv will be cyclic integral of Tds for a reversible process. That means w cycle is q cycle that is first law. But this is true for any of those cycles. I did not assume any particular cycle. So that means when I transform a loop on the pv plane which represents a reversible cycle to a loop on a TS plane which represents the corresponding reversible cycle. The area does not change. What does it mean for the transformation? It is not only area preserving but it is 1 is to 1. We say that this and that means that the Jacobian of the transformation is 1. And what do you mean by Jacobian? Have you understood what is meant by a Jacobian? See when we have let us say this is pv and this is TS. And I consider a loop here. And I consider the corresponding loop here. And I consider integral dpdv double integral over the cycle. And this area is double integral dTds again over the same cycle corresponding mapped point to point. If it were a 1 dimensional transformation and if I want the length I would have just the scale ratio coming in. Here it is a 2 dimensional transformation. So the scale ratio is known as the Jacobian of TS with respect to pv. We will define this. The scale ratio is defined the area scale ratio. This can be derived using analytical geometry. If you look up books by Thomas or Apostol you will find it there. Area scale ratio is known as the Jacobian j of say TS with respect to pv or pv with respect to TS. The other symbol used for this is partial derivative of TS divided by partial derivative of pv. And the expanded form of this is like this. It is a determinant 2 by 2. And when it is partial of TS partial of pv it means partial of t with respect to p at constant volume. Partial of t with respect to v at constant pressure, partial of s with respect to p at constant volume, partial of s with respect to v at constant pressure. This is the definition. This is the symbolism. And the name is Jacobian. Jacobian of transformation or transformation Jacobian. And it is a determinant. So if we know t as a function of p and v and s as a function of p and v we can evaluate. And because our areas are equal, this means that for us this is equal, this equals 1. If t, p and s, t, p, v and s represent temperature, pressure, volume and entropy respectively. Otherwise if you write x, y, z or p, q, r, s it is just a symbol. But we have shown that because of the first law of thermodynamics and the definition of entropy. If t is temperature, s is entropy, p is pressure and v is volume, the value of this Jacobian is 1. Now we will demonstrate that the four Maxwell's relations are hidden inside this. And for that, see one of the greatest properties of Jacobian is remember that the chain rules for partial difference are very complicated because more than one variable is involved. However, the characteristic of a Jacobian is that a Jacobian can be treated almost exactly like an ordinary derivative. And we will see what the properties of Jacobian are. So this means the transformation of Jacobian of the transformation is 1. So I have just now written this part but even the reciprocal is true. Now we will use the properties of the Jacobian for relating them. Now this is like an ordinary derivative. Jacobian of u, v with respect to x, y multiplied by Jacobian of x, y with respect to u, v is 1. So it is like saying dy by dx, ordinary derivative will be 1 divided by dx by dy. In one case you consider y as a function of x, in another case you consider x as a function of y, perfectly analogous. Not only that, the chain rule of Jacobian is also dx by dz is dx by dy, dy by dz. If x is directly a function of z that is the left hand side and directly a function of y and y directly a function of z, that is on the right hand side. So partial of that is Jacobian of pq with respect to tu is partial of pq with respect to rs, the intermediate pair of variables multiplied by partial of rs with respect to tu. But the importance is if instead of two variables in the numerator and denominator, if there is a common variable in the numerator and denominator then what happens? It turns out that the Jacobian of u, y with respect to x, y will have the same value if you just interchange the order, Jacobian of yu with respect to yx because you know the determinant does not change its value even if you flip pros and columns. But if you interchange one of them yu xy then you will get a negative sign or uy yx you will get a negative sign. But important thing is any one of them is equal to partial of u with respect to x at constant y. So all that we have to do is remember this line and you can remember it as a symbolism that partial of u with respect to x at constant y can be written down as Jacobian of uy with respect to xy. Sometimes while saying we even say partial of uy with respect to xy that means Jacobian if there are more than variable. This chain rule for Jacobians and let us see where we reach using this. Now let us say we want partial of t with respect to v at constant s. Is this a candidate for Maxwell's relation? Let me go back a bit I think I missed something. When see in thermodynamics there are so many variables, so many properties that you will come across partial of p with respect to v at constant h and something like that. We have to decide whether a particular partial derivative is a candidate in the Maxwell's relation or not. For that you notice one thing remember we remember pv diagram and ts diagram. So in our brain pv is a comfortable pair ts is a comfortable pair but pt is not pt vt or ps vs are not. Here you see one pair is t and s take the numerator variable and the constant variable. One pair is t and s another pair is v and s. On the right hand side one pair is pv another pair is something useless sv. Similarly here there is a ts pair here also there is a pv pair in some order there is a ts pair there is a pv pair there is a ts pair there is a pv pair. So when you come across a partial derivative partial of x with respect to y at constant z look at the pairs xz and yz. One of them should be pv or ts in some order then it is a candidate you will find it somewhere in the Maxwell's relation. So based on that I am considering one possible candidate which is partial of t with respect to v at constant s there is a ts so this will be somewhere found in the Maxwell's relation. So we write it first as a Jacobian s is constant so this is Jacobian of ts with respect to vs. We want to transform it so we take it and multiply it by Jacobian of pv with respect to ts. Remember what I told you about conversion factor conversion factor is dimensionless the value of conversion factor is 1. So this is an area conversion factor value is 1 dimension dimensionless so this will not change anything. So when I multiply it by 1 pv ts what happens by chain rule I can simply cancel out the ts from here and cancel out the ts from here look up the chain rule for Jacobians. So this becomes partial of pv with respect to vs and you have one variable common in the numerator and denominator that would be the common variable for the derivative this represents and the uncommon variables are not one below the other they are across a diagonal so there will be a negative sign. So this becomes minus partial of p with respect to s at constant v check out whether this is a valid Maxwell's relation you have the four relations with you check them out which one first one meanwhile you try second or third one whether you get it by this method are you getting it okay. So now we should not be scared of any partial derivative in thermodynamics find out whether it is a candidate for Maxwell's relation how will you find out take the variable in the numerator and the constant variable it could be if it is a pv ts in some order then yes if not look at the denominator variable and the constant variable it should be pv or ts if not what to do that we will see when we come to exercise and if one of them is pv or ts continue represent it in terms of a Jacobian use the Jacobian transformation because you know one of them will be pv or ts so when you use either this or its reciprocal that pv will be get will get replaced by ts and you will get the other part of the Maxwell's relation you should be absolutely comfortable with this and now remember that Maxwell's relation is nothing but mathematical depiction of what we know that if we have a reversible cycle of a fluid the area on the pv diagram which represents the work done during a cycle will equal the area on the ts diagram which represents the heat transfer during that cycle and the heat transfer during that cycle is represented by that area because it is a reversible cycle and because the cycles are reversible the directly the property relations can be used we do not have to use greater than or equal to which comes out of the second law of the model and Maxwell's relations are mathematical ghosts which come out of this so try with this and may be by today you are scared of Maxwell's relation should go away so what next the if you look up item 10 which is property relations we have come up to derivation using area relation 10.6 now I will just flip it around we will do some exercises and then we will go to the Clausius-Clapeyron relation in Clausius-Clapeyron relation we will realize what is the link with not just steam tables but the link with the change of phase and we will also be able to link the fact that remember that school boy knows two facts one the melting point of ice reduces when it is under pressure so you put a wire and hold it under ice with weight the wire goes into the ice right a school kid also knows that ice floats on water so the specific volume of ice is larger than the specific volume of water he also knows that ice requires some heat to be melted to a liquid are all these facts link to each other for example if ice some modification of ice where not to float but to sink in water would the melting point of ice instead of lowering with pressure increase with pressure these are linked and we will link them up using Clausius-Clapeyron relation.