Hi, it's Bridget at above life channel today. I have a special treat for you I'm going to use a transformative channel style, which I refer to as trance tran s channel and We are going to connect with miss Marilyn Monroe now she did advise me on my makeup and So I have a lot more than I normally would have on especially the eyes. She kept saying darken the eyes darken the eyes I'm like, okay. So I did and the red the lipsticks very much her She does not like my dress and I told her too bad. I don't have a huge Hollywood style wardrobe she would much prefer I wear black or red and Some nice diamond like or rhinestone earrings. So I just want to make sure I mention that in addition she actually I Could feel that I needed to go perfume on not essential oil Perfume and she prefers much more prefers like a cocoa Chanel or something like that like a Chanel number five Which I don't have so the kind I have it's a little too fruity for her, but we'll see how it goes She's very particular. She's got a specific taste. Let's say that All right So as always if you've watched my channel before, you know, I work with Prince in the afterlife He actually helps to manage for me Some of these other celebrities that I when I get into a trance Channeling a transformative channeling situation. So He may feel him or sense his energy here as well or I may actually speak with him while I'm transitioning into allowing Ms. Marilyn Monroe to come and be present and I can really feel her energy.  We need to just jump right in all right So clearing my own personal energy Remembering remembering that she will be in the energy space in front of my body and not actually inside my body But that the energy will flow through me through the chakras And I can utilize that and she can utilize my body gestures and stuff will respond to react to her energy She's super close Such a little body such a little body such a little body Now that feels nice actually. Thank you. I had to wait I wanted to make sure it was okay for her to Let me come in. It's a time. It's a time. I was told it wasn't a good idea to open the eyes But I really would like to oh It's difficult. I indeed I understand I understand It's all right, and you have questions for me And I'm okay with answering them the eyes are so fascinating to me such beautiful eyes. She has such beautiful eyes I want to open them so much, but oh nice cheekbones. Oh Love Love Beautiful okay, I'm ready What are your questions? So you've been kind of pushing to be To talk for the last two or three weeks and kind of wondering what that was all about Birthday coming up birthday Happy birthday to me And I've been trying for about two and a half to three weeks in your human brain times I've never been good with dates either miss Bridget. I've never been good with dates either. We have that in common However What is warm at this body just warm Some like it hot oh This is fine. I hope she likes it so we could do it again Okay, so go ahead So what surprises you most about the afterlife what's Serious questions so deep and intellectual. It's such a mystery isn't it? It's like a masquerade ball. It's like the best Hollywood night when everyone gets all dressed up and struts their Stuff and looks their best and has their diamonds out shining bright It reminds me of a night where you walk down Hollywood Boulevard in those days It was very classy and we would go to a premiere of a movie and it would be Just the night the mystery of the night and that's just something in the air seductive I would say The afterlife is like that That Celebrity motif mystique that you Tend to fall in love with her or are so curious about it's it's rather like that it It's not to be compared to in your experience with a body but in memory the best I could recall in To bring it into comparison is to like those nights in Hollywood at the movie premieres and Just the beautiful gowns and the good-looking men and the who's who That's that glamour glamour is I would say the afterlife is like Hollywood glamour and It feels a bit out of reach doesn't it for the ordinary for the ordinary one which in As a person you are we well most of us Have been ordinary and quite ordinary. I would say the afterlife is not ordinary Definitely not ordinary Does that answer your question I hope that answers your question, I'm just really not sure how to I don't really want to get too deep and You know boring for people So let's turn it the other way. What do you miss about? When you were a person dressing up Looking good. I miss the body for sure in the way in which we could decorate it adorn it with jewels and fragrance and makeup and just Incredible sophistication. It's it's really beautiful the human body is so Gorgeous, I would say I do miss that So knowing what you know now, what would you do differently with your human life? I probably would have gotten married again Or maybe not so many times Arthur is such a dear, but just really And also I'm not that into baseball, but you're was also a darling such beautiful red roses That is so Adoring isn't it? Oh the kisses. Oh, I want to say about the kisses as Bridget had watched one of these movies and said She was Interested in the kisses the kisses on my beautiful monument The stone the wall the kisses everyone kisses leaves kisses for me And so I want to give kisses back to everyone and thank you for the gorgeous This the adoration the adoring flowers and they're so beautiful and Bridget really felt that and she loved that and so That was the moment when I I was able to sneak in and asked to be invited to speak and So I want to make sure that I talk about that But what I would have done differently so much I loved so much But but I may have gotten married again It is possible had things been different. I may have fallen in love Just like in the movies and the pictures love is just Such I just loved being in love So I may have loved him or It may have committed a little better and perhaps maybe it's I Would say I may have been just a tad just a little bit hard. It's had hard to love but It's because I have so much passion so much fire in my body and so much to share with the world and There's never really enough Time or room or opportunity to really give all of that love darling Let's talk a little bit about your transition what it was like to leave your body, you know after everything else was left behind Who you may have greeted you and what it was the experience was like for you All right, I'm getting close to my wrap-up time, but I will definitely speak of this. It's not that glamorous my departure was not That glamorous, but it was dramatic the exit was a dramatic scene and I Will share that it was definitely an accident so stop Stop with all the rumors and the gossip Hollywood is known for the gossip. It was completely an accident No one caused my death. No one murdered me I would like to clear that up those poor Kennedys. They've been blamed for so much for so long and Hmm, I just want to clear that up and at the time of my death. It was quick It was like sleeping like a normal to be quite frankly honest with you I Didn't quite to realize that I was not alive anymore. I Remember being in my bed and There were I could feel sat in around me, but nothing was over my body confining me. I could feel this soft Ness And I looked I could see this light on my left side out In the corner of my room that I thought was a lamp maybe a lamp, but I think it was a Lady a woman and Now knowing what I know now it may have been an angel But she showed as a woman a helper assistant for me And Marilyn Marilyn she was speaking my name my Hollywood name, which is who I really am Let me be clear on that That is who Marilyn is who I Really am and how who I was born to be Marilyn Monroe I know there's some flutter about my name and that's Me embody me And so I would say in your words Bridget would use the words of spirit guide or and perhaps angel and She was calling my name and I could kind of see her and Things were a little fuzzy around the edges, but that's not unusual for me. That wasn't abnormal And it was so early. It was like pre-dawn hours and I Think it was about 2 a.m. Between 12 a.m. And 2 a.m. and Then at 3 30 ish 3 30 something people came or There was something that happened and I couldn't talk to them they didn't see me and At that time I was I would say that I was out of my body But I wasn't seeing my body if that makes sense. I wasn't I was more in in a different I was in the Fuzzy edges and not in the actual experience. So there was no pain at all. It was It was sort of normal like I didn't I thought and perhaps it was just a grand dream or something and it was usual to have different kinds of experiences with the the medications that I was on many medications and I would say it was just natural and then after that everything was like a Kind of like a yellow gold and I felt like I was walking a red carpet and all I could see was just the sprite yellow gold all around me and I could see Just like this path in front of me, but I would say like red carpet and The gold Posts where the velvet robes are and they were black velvet robes with gold posts And I walked and this the sprite light, but that's not unusual for me The cameras and the lights and the photographers are so much light spotlights and that's normal And so it was very comfortable. It felt warm like a hot Hollywood night and I knew that there was this Woman who was with me, you know helping me and then there was another there was a man also beside me as well And I just I guess that's the last I remember or I can recall or reflect of those moments prior to The movement into that after what you're referring to as the afterlife for me so I Couldn't hear your son in the back. Oh She's not going to be very pleased with that That's all right. The children are fine as long as they're seen and not heard The words to wrap up I want to say thank you Thank you for those who love me who continue to love me And to my Remember me For the love that I provided the love that I Felt and and try to share That's what I would she is working to come in so I may have to leave rather quickly, but I would say thank you for this opportunity And I will come again And I will talk about all the gossip you want to talk about about Hollywood if you want to I don't I'm not Beyond that It's interesting, isn't it? It's intriguing. She wants to come back really fast I can feel Prince off to my left.  She is really sweet She's got a great personality, but she's really sweet like I could feel her She's super honest about her feelings. I could feel that I have respect for that part definitely have respect for that part There was one part that I wanted to share that At first when she was off stage, we'll call it over here She kept looking to Prince like can I come in now? Can I come in now like am I on am I on like she was coming on stage and she had this soft Fuzzy sweater like I wanted to touch it. It was like a wider eye frame I wanted to touch it with pearl buttons And it was like a button up and it was like I just wanted to touch it Like it looks so soft and fuzzy and she was really excited super excited to be here and I was going to mention one thing came up and That was like a show beforehand like I mentioned my makeup and my clothing and spray and You know perfume whatever she wanted me to curl my hair and I haven't done that for a long time And I was like, yeah, I'm not going to do that for a trans channel But she really liked like I when I felt her I felt like this big head of hair Like almost like a wig with this big fluffy curly head of hair and She really wasn't all that interested in talking a lot about the afterlife She was more interested in talking about life like human life like Sharing memories and like I could feel that and relationships were really important to her I could feel that and I was seeing all these cool images of these famous people these guys too That I have annoyed these men mostly and beautiful women They're all dressed up and and I I wish I knew who they were But I did I don't know who they are like by face recognition, but she was showing me all these people and Um It's really interesting I think having just a conversation with her would be a good thing to do so that we could actually have a dialogue about Well, who is this and who is that and here's who I feel it is is that accurate kind of a thing because when it's a transformative channel I can't do that. I can't be asking her questions I have to really work hard to stay back behind the scenes and watch, you know, stay off stage and watch What's going on? So I'm interested to listen and hear back I hope you've enjoyed this trans channel. This is Bridget at above life channel Remember the purpose here is to inspire your spirit and to fill you up with hope This is why we channel the afterlife is to connect and get those Those messages get that those insights from them about our own human life experience Thank you so much for being here. Be sure to like share and subscribe to the channel So you never miss a weekly channel. Thanks for being here


So they got us some McCall here that I suppose it was 100 degrees this morning and they unplugged it luckily so we're gonna check see what's going on it was working fine earlier like it always is so gonna plug it in see what it got going on gave it the sniff sniff test it smells like food in here so let's see what we got looks like this has had a few things done to it here probably why they spent so much money on this thing so far it's like a 2011 the compressor was released yeah that's better all right at least I can see what's going on there until the start components have been changed once before the compressor was changed in 2011 pretty much everything else has been changed but it looks in good physical shape let's plug it in again just make sure it's not pumpkin it's 2000 so this thing is 20 years old and it's 115 volts so and it runs on r22 sweet so yeah it's getting a little old there at least but gotta admit this is a pretty nice cooler and it's in pretty good shape seals are in good shape I mean you can replace the components in this thing and get a lot longer life out of it I mean when it's in this good a shape I don't see any good reason to can it just buy one of the new things that are gonna fall apart even quicker so let's go and get the ante on here see what we can find out so we got that on there let's go ahead and see if we make this thing run no good place to put my camera because everything's aluminum here we go these buzz goods are gonna get it today here are giggles let's go ahead and put this on ambridge and I'll watch it with that just in case and uh we'll just push it out on our start flip it on and she's running she's pulling eight point whatever amps if you look down here it says six seven eight so right in there about the eight eight amp mark and she's running so ambridge is not going up it's actually dropping like it should so we got bad start component simple as that now here's what sucks look at the suction pressure there on that gauge so we've probably got a grand the fans are not running but uh that dropped down off a low very quickly so we probably have problems with the uh with the refrigerant charge possibly we won't know until we get the fans running um I'm just going to uh see what they want to do if they want to have me order the right ones if you want to use one of three and ones all right so we got the start components for that compressor I'm gonna get those we're gonna get those put on once we get done with that he wants us to look at a makeup air unit that doesn't seem to be heating the air for the kitchen so we're gonna look at that once we get done all right so I got my goodies up here see if we can go ahead and get some of this stuff unhooked power's been off for a while if you remember we had some issues with the pressures so we're gonna have to double check those and maybe it's something maybe it's not I get that there just like it that's good we went with oem parts stuff for the capacitor POV lead resistor here this comes all made up with the uh a little bit of a spacer and lander put this on there next as best as can be expected I'm gonna use a little piece of duct tape to hold that in place there we go not to what I preferred versus having an oem capacitor but it'll work out just fine all right so got it all in there should be able to plug this thing in and watch it start up so good so let's go ahead and kill it see if we sniff a leak out here what's kind of concerning is we got 45 50 about 50 60 pounds of suction pressure there so I hope I'm hoping it's not a restriction of some sort I don't know if this one's a txv or an orifice so go grab the leak detector I know we definitely get a lot of leaks in those freaking condensate pan all right so we just scanned all this we didn't pick up anything just uh need to scan the inside which is usually where my leaks are usually found at all right so here's a little bit better look at what we're working on here at six doors so this is definitely not a cheap unit I think they got a quote for uh eight thousand dollars something like that so these are kind of expensive for a quality one uh 66 degrees starting drop um it holds 20 ounces of refrigerant a lot of times what I'll do is I'll uh just pull the charge away it in but I'm going to uh actually just add a little bit and see what we get I'm checking subcoil and super heat and all that happy jazz generally they won't give you any of those measurements and a lot of times it won't work this one here doesn't have the uh receiver so it's going to have to stack in the uh condenser coil this comes down to some experience level that's why usually I just pull the charge and just weigh it in but I'm keeping track of what I have and so far we're like at 10 ounces out of 20 we've got a seven degree sub cooling and our head pressure is running 106 degrees and it's probably 74 in here so 84 94 104 so about 30 32 over in the end so we're probably right in that ballpark area right now super heat pretty much is not anything to brag about which don't surprise me a whole lot so boxes and that temperature and so on so we're going to let this run for a little bit but uh basically like I said this thing's got a refrigerant leak in it it's had a lot of work done to it over the years pressers txv start components you name it but the box is a really good condition because this is a cafeteria and uh so same people always working with it so it's not abused and so it lasts quite a while like I said I do not recommend this unless you have problems and so right there's your txv normally I put my probe up a little closer here but it's kind of warm in all the different spots so I do one more turn and all the quits because it's been working good up until you know they went low and start components went out but I can't help but wonder you know is there an issue with the compressor was it basically overheating which is why I'm kind of adjusting it because it just appears that uh the start components have been getting changed an awful lot and uh like every two years so it's kind of reason why I went with what I did and she just shut off before I had the chance to do it so we're right there at 35 degrees I'm going to leave it alone sometimes you dig yourself a hole we uh went a total of three total turns out I even wrote it there in case I ever need to put it back the way it was that's a big thing make sure you keep track of your turns so that's going to wrap this one up we got to go up on the roof now and check out and make a very you know all right so I think the plane was basically it was cold air this morning so we got a captive air hood and uh exhaust fan here it's not the strongest feeling in the world so let me double check the belt and it's loose as a goose so that's not helping much so let's get that fixed I should go down and grab my crowbar to pry this baby over a little tighter we'll see if we can get it with the screwdriver which ain't really the greatest way to do it I have to say a lot of our issues it's already about as tight as it can go now that I get looking at it you see here I only got about a little bit more we can go a little bit further with this thing I prefer way of doing these with the box wrench like this offset makes it a lot easier there we go that's almost the absolute maximum that can go what belt is this one it's a 4L220 it's got a nice clogged belt here it looks like it's new maybe let me try that one instead probably a used one honestly it appears it's going to work let's see if we can tighten this thing up a touch it's pretty much about about as fur as it can go yeah that's not too bad though let's see if we can get these nuts to tighten up here a little bit with the store here you know that's why you gotta look it all over they definitely got some good grease on that thing probably should have gave it a little shake made sure the bearings are all right which feels like they are see how it runs now those aren't level horizontally together doesn't look too badly off really probably needs to be a 410 belt I think I'll grab my pry bar on a level tell you what why don't we just go ahead and get on to what we were first started to do got that let's go here and see what's going on with this this thing's probably got loose belts also which would make the heat not come on because it has a pressure differential switch that's fairly tight pulley's fairly new or two not too bad I bet you maybe the limits tripped on it check the filters looks clean oh looks fairly clean let's turn this thing back on sounds like they got a damper kicking on that's not tripped maxi troll set for 70 just does not appear that it's getting a call to do its thing so I can hear let's check our pressure switches all right so our dampers are open but pressure switches didn't show as an open circuit so may double check our limit there so it doesn't look like we're getting power to our controller so I went through everything made sure my pressure switches were fine with my meter there I've got like 0.3 we're coming in through the low pressure switch and we're going out with the high pressure switch safety so basically if it's too high or it's too low it that switch keeps it from happening goes over here to the limit blah blah blah comes over here to the fire I fire I's got a neutral and a six wire which is low airflow switch I've got a hundred and twenty three volts so it appears to be we got a bad fire I control I don't have any any lights on it or anything if you look at the back of it tells you right there pilots main burners all that stuff ignition all those are outputs basically lockout alarm it's going to be an output to tell you that's tripped neutral line limits operating control I suppose we could look at number seven see if there's got power on that all right so I feel like an idiot now the burner on and off intake air thermostat I guess I was kind of thinking it was a limit switch meaning like if the burners don't run because I've had this on their bigger units this basically has to be below that temperature before it can come on which I turned it up but I didn't you can't see very well but I'm going to set that at 55 it works when I turned up to about 58 which is probably about the ball park where we're at everything worked you see the light come on there we got airflow we got control operation it's modulating so for the most part everything's working you can hear it kick in right there I checked the voltages on it and it's modulating so I'm gonna like I said set that about 55 and that should be good enough for them so anyhow that's going to wrap this one up too so I went ahead and got the belt aligned it's a lot better than what it was now so that's going to wrap this one up and until next time guys we'll catch you on the next one


There's there we are a win everybody who's here Well, if nobody is showing up as here, but I know that Lynette is here because I've been talking here The wayward child of wiki tree. Oh, that's not true. Oh, she's not wayward She's been joining on in on all the videos that we've been doing that I've seen That's cool. Happy Friday. Happy Friday. How's your Friday going? Good. They guys actually got sleep. So I'm actually awake. It's kind of nice Well, I'm jealous I'm getting a little symbol here that says that my connection is unstable. So if I disappear, I'm very sorry You'll have to drive I'll come back There's Stephen. Hi Stephen And Isabelle she's not giving her she got So I I hopped over to the discord channel for the discord server and invited everybody We've got our cozy little crew Got our cozy little crew here for our Friday night. I love that. I should have had wine wine glass It's too early. It's it's three o'clock where I am so Can't do that I'd go right to sleep That would be bad Hi Tina Hamilton Do I I think that's a new name here, isn't it? You might be right Welcome We're just killing time Giving everybody a few minutes to get in here. So we'll just be hanging loose For a few minutes. That's how we roll. It is how we roll. I think that today Once we get going here, I'm just gonna run through the instructions real quick I know everybody's probably read them, but it doesn't hurt to do it again, right? Anytime it's always good because we always have we never know like what new people will watch this That's true. It's very true You go get my screen ready to share there's Benjamin Mollsworth Hi Ben. Do you go by Ben? Is it always Benjamin? And how are we gonna have a good day tomorrow? What's that? He's already into Saturday, so I just wanted it's gonna be a good day Yes, tell the future for us Benjamin and This is Tina's first time to Friday date night glad you're here and there's Margaret Allison She's here in some background noise. I don't know if that's an echo we've got going or I have dogs I Think we have an echo going on here and says it's a good day so far yay Margaret's down under two I see Yes, Isabelle stays up late with us on Friday nights Actually, I think Isabelle is a night owl anyway. I have a feeling Yeah, that's my natural tendency it just doesn't work with the rest of the world very well Hey, there's 22 people here watching Wow Pretty date nights the past. Oh Yeah, it might be for Margaret might be from I've got my head on but Sometimes we get an echo Coming through on a one's end Always my fault she always blames me in this relationship Yes, it's always a one's fault.   I'm very sorry Julie tap dances Summer salt start wheels. Yeah It's what we do just you wait and see Well, should we pop over and take a look at the instructions real quick, let's do it. Okay I'm gonna bring up my my screen so I have our free space page which is Space wiki tree its space colon wiki tree Friday date night. It should be linked from the g2g post And we have a section there telling you what to do If you find a date Because it varies So the of course the most important thing that you can do is add the source where you found the date Once you get the date at it and a lot of times you'll be able to Determine at that point where the person is deceased if you have a birth date. That's you know like over I don't know a hundred years and when you think Yeah, it's 150 for sure you can and you can't find their date of death Go ahead and mark the date of death as you know like uncertain or non-loving or whatever that little indicator says And that I'll be sure to keep the privacy setting open so add the source and the date and Then you can scroll or I'm scrolling. I'm so tired today You can pop up to the profile menu with a profile ID menu and you search for matches to see if you have Can identify any duplicates now that you have a date And then you'll be able to propose some urges and be on your way If it turns out that the person is living again go ahead and add the source and the date before you save Add the category wiki tree admin to the profile right above the biography header So that when you do save and it goes to probably unlisted We'll know we'll know to go look at it and see what we can do with it If there is already a manager on the profile You don't need to add the category because if it's they'll still be there to take you know any trusted list requests or anything So you only want to add the category if there's no manager And when you add it to that category it just gives me and I want a list of things to look profiles to look through and we can either find more information or Find another profile manager Very rarely we'll delete them If there if there's no way to identify who the person is or where they came from or whatever Yeah Can't find a date. Oh, so sad when you can't find a date But that's why we're here Well, I said he can't look for a date.  You have a permanent date So I just linked a couple of things here Link to our help page how to estimate dates. There's just some guidelines to use out there Usually I go by I try to look at the mom's birthday if there's a mother attached and I'll add 20 years to her birth date It usually gets you in the ballpark If you only have the father I had 25 Or you can base it off of other, you know, if there's children or spouses attached There's there's guidelines out there The one thing you don't want to do is if there's already profiles Attached that have estimated dates try not to estimate off of the estimate Because then you just you're not even in the ballpark then you're you're often another county. I think So that's it. Those are our instructions and if you guys have any questions We'll be over on the discord server and the Friday date night Channel feel free to pop in We like to chat live over there you can tag me if you have a question about anything and Anybody else who's hanging out there. Well, it might be able to help as well Hey, hi Azure welcome back. She's been out for a while. We're glad to have her back And there's Charles from Georgia Another name I haven't seen here before So any questions anything to add anyone? No, I think we should let people get to dating. I Agree because that's the fun part Yeah, I'm gonna try and work on the category this afternoon while we're while we're dating You should They're piling up out there. Oh I was gonna give you our final stat too or it keeps going up and down because as time as we go through and Profiles or are opened up or you know, like if somebody's account is gone dormant and they choose to close it Then we get their profiles out thrown out into the bin so Sometimes we're above a 5000 and sometimes we're below but we are just under 5000 right now We crossed the 5000 mark about a week ago two weeks ago awesome and Then we dropped back down because we had some more profiles opened up that were probably private before So we're getting there. We are making progress. It's slow What'd you say I Didn't say anything. Oh, that was just me squeaking through your microphone Oh, oh, yes Benjamin makes a good point to just make sure that you list the date as an estimate and there is a Let me make a little banner here to show you the coding for that you can add a template that looks like this Give me a minute Like that Course you would change birth to marriage or death or whatever you're estimating usually it's birth and if you add that to the profile it throws it into another category where people can go out and Work on those two Which is also Let's see what else I saw somebody else said something oh Bev hi Bev Diaz. How do you connect to discord? Do we have that somewhere anyone? Yeah, the help page is just help colon discord Okay Yeah, I keep forgetting that you put that up there Yes, and there will be a link there Bev so you can come join us over there You'll need to get verified, but we can usually get to those pretty quickly and then you can come join us And there's Linda Peterson. Oh Yeah, she made a point. We have some a JEDCOM bug right now that is allowing some profiles to come in without dates It's not that's something that happens a lot But it does happen now and then and that's probably we've probably had a few of those come in But they're working on it. They'll get it fixed As you know, we have a very very very small team of people that do the technical work here like one person Happy there's a main discord site for wiki has a bunch of different The wiki tree challenge server is a separate server So you run it into the challenge. It's on a different server Yes Which is also fun All right Oh there as you've got the invitation code for us Thanks, Azure Let's see what else is going on Isabel it happens sometimes. Yes, we do get undated profiles not as badly as we used to She also said the small but powerful There is a yes Yes, yeah, I have it. I have a the desktop app on my computer or I have it on my iPad Have it on my phone. I'm never without just All right, let's go. Yep, let's do it. All right, you guys will see you out there.


Now, what we've done so far, this is actually known as Implicit Casting, Implicit Casting. Now, what if I want to do Explicit Casting, Explicit Casting, instead of maybe putting it up there with Person, okay Person, Person has access, it has a get name, what if I turned it into an object, again, object, alright, well, seems to error out all the stuff, because P is an object, it's not a person, and I want to stress this enough. Even though student, which is what we're doing, even though student has get name, objects, objects do not have a get name, they have a two string, they have a few other little methods here and there, but it's very generic, it has nothing. So what I can do, however, is I can do what's known as Explicit Casting, Person, New P equals Person P. So what am I doing? Well, now, even though I know student is a student, right now it's being treated very generically like an object, I need to at least tell it where in sort of the hierarchy to act like, and so now what I've done is I've said, take this object, turn it into a person, give me that person's name. We compile this up again, New P, sorry, New P, and we, hold on, there we are. We compile this up, notice there was no errors. So what I'm saying is this very generic thing, object, I want to turn it into a person. I want to force it. This is no different than if I were to say int x equals int 3.14, we're doing the exact same thing, relatively. So now when I run this, Mr. Adam, because we have that method now, we have that access. If I were to get rid of say this override over here, just because I want to demonstrate it, I compile it again, Java C, Java, Adam. If I did student, and I said to just do my two string with student, the same thing happens.


Good morning. So, in this lecture, we will study the phase curve and that will be our first lesson on vector analysis vector calculus. In this, I would advise that before going through the lecture, you should ensure that the basic notions of vector algebra are firmly in your mind, so that there is no difficulty in following through the discussion that we do in vector calculus. So, the text book, the first section of this chapter of the text book, recapitulate the basic notions of the vector algebraic relationship. We will not go through the details of that, but I advise that you go through those things before following this particular lecture. It should be clear the, what are the implications of dot and cross products in vector algebra and scalar and vector triple product because they will be very frequently used in this lesson and the next. Further, the differentiation rules of vector functions should be clear in following the analysis that we take up now. A few points are special which I would like to remind you. One is that when we try to interface the typical vector algebraic notions with the matrix algebra, then we find that the dot product in between two vectors is equivalent to this A transpose x, which is the inner product in vectors as understood in the linear algebra terminology. Next, quite often you will find that you come across expressions of this kind A dot x, B in which A, B and x all three are vectors. So, in that situation, if you take the expressions of A dot x as it is and put that in this place, you will find that the components of x, that is x 1, x 2, x 3, they get somewhere lost in the interior of this expression and sometimes it happens that you want to treat x as the right side of a linear algebra expression. In such situations, you can consider B to be kept on this side and it does not matter, because A dot x is a scalar and if you do that, if you keep B on this side and then you have an expression which looks like B A dot x and here in this A dot x, if you use this relationship that is A transpose x, then you will find that then after that you have all three linear algebraic notions B is a vector, column vector, A transpose is a row vector and then x is again a column vector and then using the associativity of matrix multiplications, you come to this relationship in which B A transpose multiplied together gives you a 3 by 3 matrix and x is a 3 by 1 vector and you find that you have got the vector x on the right side and that may be helpful in situations where you want to treat x as the unknown or variable vector with A and B as constant and known vectors. Another important interfacing between the vector algebra and matrix algebra is in the matter of cross product. Cross product is something which unlike the dot or inner product is a matter which is very special to three dimensional vectors. For general n dimensional vectors, you do not have anything called a cross product. Now, in the case of three dimensional vectors, which are sensible in geometry ordinary Euclidean spatial geometry, you have got the notion of a cross product between two vectors and in 3D space, it makes direct sense and the result is a vector. In 2D space also in certain circumstances, there is an interpretation which makes sense. So, for example, in the x y plane, if you have two vectors A and x, then their cross product is a vector which is actually perpendicular to the plane of A and x and that means perpendicular to the plane under consideration and that way, if you take the straight away expression from a cross x, then you get a scalar value. So, in the vector algebra sense, whatever you get as A cross x, if you want to put it in the linear algebra sense, then you can do it by this relationship. In the case of 2D vector, you get a perpendicular transpose x.  So, if the vector A is A x A y, then the rotated vector rotated by 90 degree is this vector and inner product of this vector with x gives you the same result as you would get as A cross x. In the case of 3D vectors, if A and x are 3D vectors, then A cross x can be represented through this matrix vector multiplication, where A tilde is a 3 by 3 matrix which is a skew symmetric matrix with elements of A A, that is A x A y A z arranged in this manner. So, quite often in simplifying complicated expressions involving vector algebra notions into linear algebra formulation, these relationships will help you. Now, with this background of the basic notions of vector algebra rules, we proceed to the study of parametric curves or curves in their parametric representation. When we try to describe a curve in space, we can talk of several kinds of representation. This is called the explicit equation of a curve in which y and z coordinates are expressed as functions of x coordinates. As you know that for all curves, this may not be possible. That is for a general curve, it may not be possible. It will be possible only when for any given x, the values of y and z are unique. So, if a curve goes like this, then with a given value of x, there can be several values of y and z which will be on the curve and such a curve will not be expressible in this kind of a representation. This representation through two equations in x y z like f of x y z equal to 0 and g of x y z equal to 0 is another representation of a curve and this is called the implicit equation. In this case, the idea is to represent a curve as an intersection of two surfaces. f of x y z equal to 0 is the equation of one surface and g of x y z equal to 0 is the representation of another surface and the intersection of these two surfaces is a curve and that is the curve which is expressed through this relationship. Now, this is a representation which is actually unnecessarily complicated. For the description of a curve, the prior description of two surfaces is going to be much more complicated than the original issue. In comparison to these two notions, these two representations, you will find that the parametric representation is much more straightforward and completely general. In that, the position vector of a general point in a curve is represented as a vector function of a parameter. In this case, t and in terms of t, you can express all the three x y and z components as x of t, y of t, z of t and when you combine them together, you get this as the representation of the curve. So, as the parameter t changes, you find that the x y z coordinates of the point on the curve changes and accordingly, if you try to plot, you get the curve. You can represent this curve in this manner or in this manner. The two representations are equivalent. Now, if you represent a curve in this kind of a parametric representation, then a few notions appear immediately. If you differentiate this vector function of the scalar variables, then you get this derivative and this is called the tangent vector. So, if this is the curve at this point, suppose for a value of t, value of the parameter t, you get this point, say origin is here, then by the standard definitions of derivative, you get the derivative of this position vector at this point as a vector like this. This is the tangent vector to the curve at this point and this is r prime. Now, if you take the modulus, the magnitude of this vector, then what you get is called the speed of the curve, speed in what sense as if the parameter t is time, then you will find that r prime gives you the velocity of a particle moving along this curve with time t as the parameter. So, if you consider the parameter t to be time, then what you get as speed is indeed the physical speed that we know of. If you consider the unit vector along this direction, then you get the unit tangent as this vector divided by its magnitude. In this manner, if you find out the derivative, then you can use it to find the length of a curve from one point to another. Suppose, this is the point corresponding to the value of t as a and this is another point which corresponds to the value of t equal to b. Now, from here to here, if you want to find out the length of the curve, then what you can do? The speed at which it moves with respect to t at this point. So, you take that and multiply that with a little small change d t, then you get this segment. Similarly, you get lots of segments. So, the sum of these segments will give you the length of the curve as the length of each of the segment tends to 0 and there are infinite such segments, very large number of segments and each segment extremely small. That means, it is the sum of a large number of small parts that is integral. So, you get the length of the curve from t equal to a to t equal to b as this in which you have d r mod here, which means d r by d t and d t separately. So, d t is here and d r by d t modulus is this. So, this gives you the length of the curve. Now, if you keep the initial point for calculating the length as constant and the final point, if you do not specify as a particular value, but if you keep that as variable, then through that exercise, you can define a very important function, which is called the arc length function. So, if you keep b variable, then you get this arc length function, which is the integral from a to t. That means, starting from this point, for every value of t, you get an arc length, which is the length of the curve starting from this point to the current point. So, this is a function of t and this is called the arc length function in which you get d s is equal to the same mod of d r, which is this and from here, if you represent them as d x by d t whole square plus d y by d t whole square plus d z by d t whole square, then a term d t comes outside and here, what you are left with is simply d s by d t. That is this. Now, a curve r of t is called regular, if the tangent vector never vanishes. That is at every, if a curve is called regular, if at every point of it, there is a non 0 tangent vector. The direction is clearly representable from the derivative, from the first derivative and in such situations, you can effect a re parameterization of the same curve with respect to some other parameter say t star, which must be a strictly increasing function of the current parameter, which is t. Now, if you check the arc length function for a possible parameter, then first you note that arc length s of t is obviously, a monotonically increasing function. That is as t changes as t increases along the curve, you keep on getting the values of s, which is continuously increasing. So, arc length is a monotonically increasing function. Next for a regular curve, if r prime is not 0, then d s by d t will not be 0. And in that case, the function, arc length function s of t has an inverse. That is, its inverse can be expressed as t of s. And if we can do that, then wherever we had the representation of the function as r of t, we can revise it as r of t of s. That is, by virtue of the fact that the arc length function has an inverse. That is, from s of t, we can derive the inverse t of s. And if we can derive the inverse, then we can insert that inverse here. And then finally, this becomes a function of s. So, in this manner, we can define the curve as with s as the parameter, with the arc length itself as the parameter.  If we parameterize a curve with the arc length as its parameter, then we get something interesting. What we get in that case is that, if we represent it as r s, then as we try to find out the derivative for the tangent, then we get r prime s. And this will be always of unit length. This will be always of unit length. And that is why a curve parameterized with arc length as the parameter is called a unit speed curve. Because for every small distance covered, you find that the length of the small segment is same as d s. So, this derivative will be always of unit length. So, the tangent that you will get will automatically have unit length. So, you do not have to really divide it with r prime norm. r prime norm is 1. So, in that case, that unit tangent you get directly as u of s as r prime s. Now, we come to the most important property of a curve, that is the curvature. After defining the tangent from the first derivative, you try to take the second derivative.  And now with the arc length parameterization you have got the unit tangent. The ordinary tangent turns out to be a unit tangent directly. And in that case, all that you need to see is that how this unit vector changes as you proceed along the curve. And that gives you the notion of curvature. In undergraduate calculus for planar curve, you might have studied this notion of curvature. That is, curvature is d psi by d s in which psi is the angle that the tangent makes with the x axis. So, here also you find the notion is same. That is, as a particle proceeds along the curve, then with respect to distance covered along the curve, how fast the direction changes, direction is represented by psi. The same thing is here, in which we say that with respect to s, how fast this tangent vector u is turning. So, we try to take the derivative of u with respect to s. So, d u by d s. And that gives you the measure of the curvature. And the way this curvature function is defined, it is going to be non-negative always. Because we are defining the curvature function with a norm, which will be never negative. So, you can say kappa of s, that is curvature function with respect to the arc length is norm of u prime or norm of r double prime. So, from here as you do that, you note that the unit tangent, if this is the u, u of s in which case, we will be defining the curve not with respect to t, but with respect to s. Then, if this is u s, then note that this is a unit vector. And with respect to the arc length, the rate of change of the unit vector will be perpendicular to this vector itself. Because for any small segment, here it is a unit vector, here also at the next point also it will be a unit vector. So, if you take this as u, then at the next point also it will be a unit vector, say u plus delta u. And then, this is the change delta u. And as s tends to 0, as this change becomes very small, then you will find that this change of u, that is delta u, turns out to be perpendicular to u. So, any unit vector you take and you consider its derivative with respect to arc length itself, then you find that the derivative is perpendicular to the unit vector. So, in this case, that perpendicular vector will be in this direction, the way it is defined, the vector like this will be always in the interior, that is in the direction in which the turn is taken by the curve. It could be 0, if around that place, around that location, temporarily the curve was going straight, then it would be s 0 vector. Otherwise, if there is a turn, then it will be in towards the interior of the turning. So, this vector is a normal, normal vector to the curve at this point. And as we know that its magnitude is defined as kappa. So, its direction, the unit vector along this direction, which is found by dividing u prime with kappa, that is called the unit principle normal. Unit, we can understand, because this is going to be a unit vector, because we are dividing u prime with its own magnitude. So, it will be a unit. Normal, we can see, because it is at right angles to the tangent vector. So, that is normal to the curve at this point. But, why do we call it unit principle normal? Why principle? Because, at this point, if this direction gives us the tangent, then perpendicular to this line, at this point, we can actually draw a plane. And every vector in that plane passing through this point is actually a normal. And this is the principle normal. Other vectors in this plane are also normal, but they are not the principle normal. This vector is called the principle normal, because it, this vector along with the tangent defines the plane, which can be taken as the plane of the curve at least locally, at least around this point. That is why the normal, which falls in the plane of the curve in this local neighborhood, that is the principle normal. If you take the cross product of u and p, if you take the cross product of u and p, then you get another normal out of the blackboard, out of the plane of the blackboard, and that is called the binormal. Now, this is easy for a representation with respect to arc length as the parameter. In the case of a general parameterization, in which we use the parameter t, not the arc length, for this curve, you will find that it will not be so easy to derive the expressions for the principle normal and the curvature function. So, for that, what you can do is, you can consider it in this manner, that r prime t has a magnitude, which is not unity in general. So, you can say that r prime is its magnitude, not necessarily 1 into the direction, unit vector along the tangent. Now, if you differentiate this for developing r double prime, then you will find that it will have two parts, one because of the variation of this and the second because of the variation of this. In the case of r of s, this part was missing because it was constant. So, if you consider a general parameterization like this, then you get r prime, which can be represented in this manner, and then if you formally differentiate it using the product rule, then you will get the second derivative in this manner, that is the derivative of r prime norm into u t plus r prime norm into d u by d t.   So, when you insert those things, then you get this kappa into p and d s by d t gives you one r prime norm and another r prime norm is already sitting here. So, in the second term, you get this, first term has remained unchanged. Now, you find that this second derivative is not a vector in the direction of the unit principle normal, but it is a combination of u and p. The reason is that here the speed is not constant, speed is changing. So, you have this. So, because of the change of speed, you get this is acceleration actually. You get one component along the tangent, that is because of the change of speed, change of the magnitude of speed. And this is the representation of the part which is the result of the change in direction. So, now from here, if you want to find out the curvature, then what you need to do? From this second derivative, you need to subtract this part out. What remains? You examine that and from there, you get the direction vector p t and the curvature, because you have already r prime. So, when you consider r double prime minus this term, whatever is its component along the tangent, that part you subtract, then whatever is left from that, you get the direction and by inserting the values of r prime, you get from the magnitude, you get the curvature. Now, this is the situation. If you try to see it schematically, then at point a, this is the position vector r, r of t. That means at every point, r of t changes. And at this point, this is say r prime, r prime of t, unit vector size u, size 1, this is the unit vector u, u of t, this and then as you calculate the secondary derivative of r, it is not necessary that it is normal to the curve. It may be in some direction like this. Out of that, when you subtract away the tangential component, then what is left is in this direction and in this direction you get this vector. The unit vector along that direction is shown as p of t and whatever is kappa, that is the curvature. If you take 1 by kappa, then you get a length and in this direction, if you mark out that length, then you get a point denoted by c here in this figure and that is called the center of curvature, y center of curvature. Because locally around this point, in that case, you can represent the very close neighborhood of the curve by a circular arc, the center of which is at p. So, in that sense, this is the, this is called the center of curvature. And the plane that is defined by these two vectors u and p, that is the plane in which the curve resides in the immediate neighborhood of this point and that plane is called the oscillating plane. As we have already discussed that the two principle unit vectors u and p, if you take from the oscillating plane and define this cross product, then you get the third mutually perpendicular vector b like this perpendicular out of the plane of the blackboard. Then that vector is called the binormal. Why binormal? Because it is normal to the curve certainly and it is normal to both the important vectors that we have earlier defined u and p, that is why it is called the binormal. Now, you find that through derivatives, we have defined three unit vectors at this point which are mutually perpendicular u, p, b and u, p, b together define a right handed triad, the kind of three vectors that we typically use to represent vectors to get a frame of reference to describe 3D vectors. So, this particular right handed triad formed by that point as the origin, this particular point as the origin and these three vectors, these three unit vectors u, p, b as the three reference vectors, that frame of reference is called the serrate frenet frame. Now, in this frame, you can directly describe the oscillating plane which is the plane formed by u and p and then if you consider the plane formed by u and b, that is called the rectifying plane and the plane formed by p and b, the two normals that is called the normal plane because that entire plane formed by p and b is normal to the curve. After this, we get into the discussion of the next important property of a curve which will be there in the case of spatial curve, three dimensional curves. In the three dimensional curves, the tendency of the curve to twist out of this local plane, the oscillating plane to the tendency to twist out of this plane and become a spatial curve go out of this plane is measured by the quantity torsion. So, how would you describe it? You find that the plane, oscillating plane is the plane formed by u and p and the direction of the plane is given by the direction of it normal which is the binormal b in this case. Now, as the curve tends to come out of this plane, then at the next point, the oscillating plane will be somewhat different which means the binormal will be somewhat different. So, then the way the curve tends to twist out of the oscillating plane will be measured by the way this binormal changes that is from this unit vector, if it changes to this unit vector, then the manner in which this change takes place will give you a measure of the twisting tendency of the curve or torsion of the curve. That is why we measure torsion through the rate of change of b with respect to arc length.  Now, here u prime we already know as kappa of s into p. So, we insert that and then we find that the first term has p cross p which is 0. So, this goes out. What remains? You have u cross p prime. Now, we know u p b, now we ask what is p prime? Now, what is p prime? Till now we have not come across the expression for p prime, but one issue is very clear p is a unit vector and therefore, its derivative with respect to the arc length must be perpendicular to p itself. So, if there is a vector which is perpendicular to p, that means it will be in the plane of u and b. So, we can consider p prime as a linear combination of u and b. So, let us take it like this and try to put it here. So, this kind of an expression with a linear combination of u and b sigma u plus tau b we insert here. As we do that, we find u cross this. Now, the first term in this product will involve u cross u which is 0 and the second term will be tau into u cross b.  So, you will get this. See, we have got b prime the rate of change of the binaural in terms of the known vector. So, it is a vector in the direction of the principal normal p and the magnitude tau that is we find here is called the torsion of the curve. And how do you find it? You just take inner product of this equation with p and then you get p dot b prime negative of that as tau. So, from here we get first the torsion of the curve, second the derivative of the binormal which is in this manner a vector in the direction of p itself. Now, you see that earlier we defined u prime and we got that as kappa into p when the parameterization is with respect to the arc length. Now, we have defined and got an expression for b prime. Now, out of the three vectors u p and b for two of them for u and b we know we have described how they change that is we have evaluated their rate of change u prime and b prime. Now, what about p prime? Here we assumed that p prime is this linear combination of u and b because we knew that it must be in the plane of u and b, but this sigma and tau we did not determine at that time tau turns out to be the torsion of the curve and that is how it is defined. Now, sigma is left if we can determine the value of sigma then we would have completed the description of p prime that is the rate of change of p that is how this principle normal changes. So, we can do that now completely because we have got now both u prime and b prime and from this triad we find that p is b cross u. So, using that we can try to determine p prime from this expression itself. So, if p is b cross u then its derivative will be b prime cross u plus b cross u prime and if we use the expression for b prime as we derived just now. That is b prime is minus tau p if we use that here then we find minus tau p cross u plus b cross u prime which is kappa p. Now, if we simplify this we find that p cross u is minus b.  So, we get from here minus kappa u. So, the sigma that we assumed here turns out to be minus kappa and with this now we have got p prime also. This complete set of three formulae that give the rate of change of the three unit vectors u p b in terms of those vectors themselves are called the serrate finite formulae. Now, with these formulae if you know the curvature and torsion as functions of the arc length then given the initial vectors u p b at the starting point of the curve you can determine the curve for all subsequent values of p. How do you do that? If you know not all subsequent values of t, but s the arc length function. If you know kappa and tau as functions of s then with the given value of u p and b at s equal to 0 you know the complete right hand side. That means you know the rate of change of these three vectors u p b. So, the initial trade is given and from these formulae you can find out the rate at which they change that is for a very small distance d s travelled along the curve you know how these will change the d u d p and d b you can work out. And with that with that knowledge you can effect that little changes in d in u p and b and proceed to the next point along the curve. As you know that next point along the curve and the u p b values u p b vectors at that point and again you know kappa and tau as function of s then at that new point you can again completely determine the right hand side and keep on determining the next point and the next trade u p b next frame or the next serrate frenet frame. And this way you can continue along the entire curve not only that if you have two different parameterizations of a curve starting with different points then also, but if you find that they are the same curve, but separated one curve here starting from this point and going like this and another curve starting from here going like this. If they are the same curve then with these two different starting points and different directions in which you proceed you can find out that these are actually the same curve moved and rotated by this distance and some rotation. So, this you can establish if you can find out what are the u p b vectors at this point and what are the corresponding u p b vectors at this point. And if you can give a compensatory motion to this compensatory rigid body motion a rotation and a translation then you find that the entire curve comes back at this location. So, that means that if the curvature and torsion kappa s and tau s are two given functions that is curvature and torsion as functions of arc length parameterization as long as these two functions are same for two curves the two curves will turn out to be exactly the same curve except that they may be rotated and displaced through some rotation and a displacement. So, that is why we can say that the serrat finite frame and the curvature and torsion functions give you an intrinsic representation of a curve intrinsic representation in the sense that they do not depend upon an external frame of reference. The u p b triad itself is its own most natural frame of reference in which to describe the curve and that frame keeps on moving along the curve as you change the parameter value that is the arc length. So, this result you can establish very easily that is the arc length parameterization of a curve is completely determined by its curvature function and torsion function except for a rigid body motion. Some of the exercises in the in the corresponding chapter of this of the text book you will find two different looking functions two different looking vector functions representing the same curve and through the reduction to the standard form through arc length parameterization you can establish the equivalence of the two different parametric representation for the same curve located at different locations. With this much discussion on curves we conclude this lesson with a little discussion on parametric surfaces. The way we describe curves by the help of a parameter we can also represent a surface by the help of two parameters say u and v. If you can represent the position vector of a point on the surface with the help of two parameters as a function of two parameters then the representation looks like this x of u v i plus y of u v j plus z of u v k. You can also represent it as a column vector with x y z functions of u and v being the members meaning the components in the vector. Now, for a function for a surface which is a two dimensional entity at every point you can find out two independent tangent vectors that is you can define a complete tangent plane and on that tangent plane you can define infinite tangent vectors two of them will be linearly independent in terms of which as linear combinations of which you can represent all the other tangent vectors. So, for example, suppose this is a surface then at every point of it you will find that you can describe a complete plane which is tangential to it. So, from this point you can determine several tangent vectors like this. So, in that plane all vectors in that plane which are touching the surface. For example, if you draw a cylinder like this then at this point you will find that you can find out a complete plane every point of it is a tangent. So, the line like this is also a tangent and in that plane whatever in the plane perpendicular to the board you can draw this kind of a tangent this kind of a tangent and so on. So, in that tangent plane two of the tangent are linearly independent as a linear combination of these two you can construct any other tangent in that same plane.  Now, if the parameterization is not degenerate at that particular point then you will find that r u and r v are linearly independent. There may be a point there may be some points where the two parametric tangent vectors turn out to be in the same direction and that will be a degenerate point for this particular parameterization. Now, considering a situation in which the point that we are considering is not degenerate you can find two linearly independent tangent vectors r u and r v by straightforward derivative. So, as you find two such vectors two such a tangents in the tangent plane then the cross product of these two tangent vectors will give you the normal to the surface like this which is normal to the entire tangent plane. So, that normal you can represent like this and quite often you are interested in the unit normal vector which you can find out like this. Now, the local shape of the surface around this point can be explored can be analyzed through the investigation of how this particular normal this unit normal changes as you move along the surface along this tangent or that tangent or any other tangent. So, if you have got say a surface like this at this point you have got this as r u and this is a tangent vector and this is another tangent vector this is r v cross product of them gives you this normal. Then at this point if you say that now along r u if I try to move on the surface then I get a point which is arbitrarily close to it close to this and a little in this in the direction of r u along the tangent plane then you move to this point which is very close to the original point here and what is the normal at this point. So, that normal you can work out similarly if you move along r v then you can get another normal and then you try to see the projection of these normal on the tangent plane again. So, that way for every move along a tangent vector you can get a new normal and you work out the rate of change of this unit normal unit normal n and through the those rates of changes you can work out the local shape of the surface around this point. So, how does you ask the question how does n vary over the surface. So, this gives you the information about the local shape local geometry of the surface. So, you can determine a curvature which is called normal curvature at every point along every direction. So, in the tangent plane if you take another direction another independent direction neither r u nor r v some any other direction arbitrary direction in the tangent plane which is a linear combination of r u and r v then you can cut the surface through a plane which includes n and this vector the normal and this vector. So, you can make a normal cut on the surface through the normal plane of n and this vector and that will define a planar curve and the curvature of that planar curve is called the normal curvature of the surface along this direction at this point. And then that is another candidate for a direction for the tangent and then if you can find try to if you try to find out those directions along which you can determine the normal curvature in such a manner that the curvature tensor that you get you see that curvature is actually a tensor because the moment you decide on a direction in the tangent plane finally, you get another vector in the tangent plane itself. So, from tangent plane to tangent plane that is a mapping. So, that way the curvature at every point is actually a tensor quantity and as you diagonalize the tensor quantity the curvature tensor you can find out those directions in which the entire curvature is contained within the normal plane. So, those directions are called principal directions and the corresponding normal curvatures are called principal curvatures. So, you can diagonalize this curvature tensor and find out two principal directions and the corresponding principal curvatures depending upon the signs of these principal curvatures you can determine you can classify the local shape as convex, one cave, saddle and so on. So, if both the principal curvatures turn out to be positive then you say that locally the surface is convex. If both of them are negative then locally the surface is concave. If one of the principal curvatures is positive and the other is negative then that point is termed as at that point the surface geometry is termed as a saddle point that is a saddle point. If one of the principal curvatures turns out to be 0 and the other is positive or negative then it is cylindrical. On the other hand if both the curvatures both the principal curvatures at a point turn out to be 0 then locally the surface the surface is a planar surface. So, this way you can work out the local geometry of a surface based on the way the normal varies in the immediate neighborhood. So, let us summarize the important points in this lesson. First point is that parametric equation is the general and most convenient representation of curves and surfaces in three dimensions. And second for curves for parametric curves arc length turn out to be the most natural parameter and the saddle turn out frame gives the most natural description of the curve which is intrinsic to the curve which does not depend on any external reference. Third important point is that the curvature and torsion are the only inherent properties of a curve. The tangents etcetera are themes or properties which are dependent on frame of reference also parameterization also and so on. Curvature and torsion as functions of the arc length parameter are the only intrinsic properties of a curve the rest of it is due to parameterization and frame of reference chosen. Finally for a surface patch for a surface patch the local shape can be understood through an analysis of the curvature tensor which is found by exploration of how the surface normal changes as we move in the immediate vicinity of the surface around a given point. Now, in the next lesson we continue the vector calculus topic and in this lecture we will see some of the differential operators on field functions. And in the next lecture we will continue into a discussion of the integral operations and integral theorems. First the differential operators a function of x y z which is a scalar function that is a mapping from three dimensional space to the real line is called a scalar point function or scalar field. Similarly, if you have a mapping from three dimensional space to a three dimensional space that is a vector field or a vector point function you can represent it with this in this manner. Now, on both of the scalar and vector fields you can apply an operator the differential operator first order differential operator which is denoted as del or nabla and the meaning of which is this. Note the particular way this particular this operator is written normally when we write a vector with components along i j k we write a x i plus a y j plus a z k, but here it is written as i into something plus j into something plus k into something the reason is that this something is not a quantity, but an operator. So, when you carry out the algebra of this del or nabla operator there may be situations where slight changes will mean different things. So, to save oneself from confusion one must remember these three things one is that del is a vector quantity second it signifies a differentiation and third it operates from the left side. You cannot multiply this thing to a quantity from the right side and expect that you will get a quantity back you always apply del from the left side.  So, you can think of dot problem between them you can think of this and you can think of this, but both of them are meaningful expressions, but they mean two completely different things.  What is this if you apply the del operator through a dot product then as you apply that on this on v and you work out all the dot product as usual then you will get something which will be del v x by del x plus del v y plus del y plus del v z by del z which is a quantity which is scalar quantity and this is this has a meaning this is called the divergence of v as we will shortly see. On the other hand this one will mean v x del by del x plus v y del by del y plus v z del by del z. Now, this is also scalar, but this is not a quantity this still is an operator waiting to operate on some function on this side. So, these two mean two completely different things because this del is an operator and it operates from the left side. So, when on the right of del there is nothing given. So, it remains still an operator after we put something on the right side of it then this operator will operate from the left side and produce a quantity. Now, another important operator which is a second order operator for that matter is this del square which is this. This is called the Laplace-Tien operator because when this operator is operated on a function phi then you get this equation which is the famous Laplace's equation and a solution of this kind of an equation that is a function phi which satisfies this equation is called a harmonic function which we will come across quite often in the course particularly in this chapter also. Now, when you apply the first order differential operator del on a field function scalar field or vector field you can develop three different notions of vector calculus. First is the gradient which is the result when you apply the del operator on a scalar function a potential function. So, that is the gradient quite often denoted as grad phi or simply del phi like this. If you apply that del operator term by term on phi then you get this and this is the gradient vector. This gradient has the same meaning as the gradient which we discussed in the previous lecture. That was in the in that case the dimension was free in this case the dimension is just free. In that in the other case in the linear algebra sense the dimension could be anything it was free here it is only for three dimensional vectors that is why three components i j k are there. So, now if phi of x y z is a scalar field then phi of x y z equal to constant will give the level surfaces or equipotential surfaces of that potential function and at any point the corresponding gradient vector that you can determine like this turns out to be normal to that or so on and to the level surfaces that is the gradient. Now in the scientific problems in engineering problems this is very important because if in a potential flow you have the potential described by a function phi then the negative of the gradient phi gives you the velocity vector. Now on a vector function when you try to apply the del operator then del itself being a vector on a vector function when you try to operate it you can operate in two manner one is by a dot product and the other is by a cross product. So, through the application of a dot product between the del operator and v you get this kind of a thing and which is called the divergence of the vector field or divergence of the vector point function. So, the vector field is given like this v x v y z are its components along x y z directions and the divergence is given by this expression. Now this also has a direct meaning in the context of fluid flow that is divergence of rho v where rho is the density and v is the velocity vector gives us the flow rate of mass per unit volume out of a given control volume. There are similar relationships between field vector and the flux in the case of electricity and magnetism. Now if rather than dot product between the operator del and the vector function v if you have a cross product then as a result you get a vector function because cross product between two vectors is a vector and that is called the curl and that is defined in this manner curl of v is like this cross product which you can evaluate like this and get the three i j a k components in this manner. So, that word curl literally means rotation turning and that is the precise meaning when you try to explore the situation in the case of a fluid flow. For example, if v equal to omega cross r represents the velocity field then the curl of the velocity vector gives twice the angular velocity that is twice the rotationality. So, that means if curl of the velocity vector is 0 that means omega is 0 and that means it is an irrotational flow. So, curl represents the rotationality in the flow. In electromagnetism the relationships between electric field and magnetic field is also established through the curl of the corresponding field. Beyond this we will consider a few further differential operators, composite operators and second order operators and integral operators in the next lecture. Thank you. .


Here we are at the Southern California Earthquake Data Center catalog search page. What we're about to do is create a catalog that is the same length in time as our one-year New Matrid catalog and also covers the same latitude-longitude box area. So that way we can really compare how active this area is in terms of number of earthquakes produced compared to New Matrid because we'll be looking at the same size and length of time. Alright, so the catalog to search is this one. The output format default is fine. Let's make a one-year catalog for the same year as we did before. So let's make it 2011, day one, month one, like that. We'll leave the magnitude and depth defaults alone. For latitude, let's change it to 33 and 38. And for longitude, we're going to change it to West 124 and West 119. I'm going to shift click to select local regional and tall sizes. And I want to send this output to a file. So you're going to save this as plain text somewhere on your computer so that you can manipulate it to make a frequency magnitude plot later on. Let's get rid of that right now. Now I also want you to make a map. So in that case, what you do is go back up here to the output format, leaving everything else the same and change the output to a Google map. And now you're going to want to output this to a web page, submit request, stare off into space while this thing happens, and you have a map. You might want to zoom out or center it differently. And then take a screenshot of this map and paste it into your problem set.


Everyone were at Senator Murphy's office, Senator Warren's office, Senator Shat's office. Senator Durban, because yesterday or early this morning he voted yes to send billions more money and weapons to the Israeli government. He just killed more than 28,000 people, injured more than 60,000 and destroyed more than 70,000 units in Gaza. He voted to permanently end U.S. funding for UNRWA, humanitarian aid, and people in Gaza are already dying from lack of food, water, and medical supplies. As a Palestinian, as an Ohioan who has great love for Ohio, I'm disappointed, does not even begin to describe how we're feeling right now. The Muslim population, the Palestinian population, the American population is disappointed in this. If you were a Palestinian, would you feel safer from this legislation? Knowing that Israel will have more money to drop bombs and UNRWA has lost their funding, that would not make me feel safer as a Palestinian. There's no compromise on genocide. We want to leave this with her, wanted complicity in Israeli war crimes, Senator Warren.


So Deegan, watch how I do a bridge. I am going to start lifting myself up. Notice how I hold my bottom off the ground like a bridge when I go up. Rigiditch, it's a bridge. Make sure you hold your bottom in the air like a bridge. Now see if you can do 10 with me. Rigiditch, it's a bridge.


Hi and welcome to the session. I am Shashi and I am going to help you with the following question. Question says verify Rolls theorem for the following function. Given function is fx is equal to x square minus 5x plus 4 on closed interval 1 comma 4. First of all let us recall Rolls theorem. Rolls theorem states that if function f from closed interval a comma b to r is continuous on closed interval a comma b and differentiable on open interval a comma b such that f a is equal to f b then there exists some c in open interval a comma b such that f dash c is equal to 0. This theorem is the key idea to solve the given question. Let us now start with the solution. Now we have fx is equal to x square minus 5x plus 4. Clearly we can see this is a polynomial function and we know every polynomial function is continuous. So we can write fx being a polynomial function is continuous in closed interval 1 comma 4. Also given polynomial function is differentiable in open interval 1 comma 4. Now using key idea we will find values of f1 and f4 and we will check if f1 is equal to f4. So first of all we will find f1 it is equal to 1 minus 5 plus 4. Substituting 1 for x in this expression we get 1 minus 5 plus 4 which is further equal to 0. Similarly we can find f4 substituting 4 for x in this expression we get 4 square minus 5 multiplied by 4 plus 4 it is equal to 0 only. Now clearly we can see f1 is equal to f4 so we can write f1 is equal to f4. So clearly we can see all the conditions of Roll's theorem are satisfied. So we can say there exists some c in open interval 1 comma 4 such that f dash c is equal to 0. Now we can write so there exists some c in open interval 1 comma 4 such that f dash c is equal to 0. Now f dash c is equal to 0 implies 2c minus 5 is equal to 0. We know fx is equal to x square minus 5x plus 4. Now differentiating both the sides with respect to x we get 2x minus 5 is equal to f dash x. So we can find f dash c by replacing x by c in this expression and we get f dash c is equal to 2c minus 5. Now this further implies 2c is equal to 5 adding 5 on both the sides of this equation we get 2c is equal to 5. Now dividing both the sides of this equation by 2 we get c is equal to 5 upon 2. Now 5 upon 2 belongs to open interval 1 comma 4. So we get value of c is equal to 5 upon 2 and this value belongs to the open interval 1 comma 4. So we can say hence the Roll's theorem is verified. This completes the session hope you understood the solution take care and have a nice day.