noc19-ma19-lec01

[Music] [Music] welcome to the second part of the promise course on calculus of more than one variable I'm happy that the course on calculus of one really evil has been well-received it has had one rerun or two reruns I guess maybe you'd have a rerun in the current when next session of the MOOC courses as before I would like to concentrate on the board but keeping in view that there has been a request to be on doing things on the tablet but I believe that just being in the tablet is a quite a boring thing you would be just seeing somebody doing blah blah blah and then something appearing here on the screen unless some drama is done on the blackboard math doesn't really becomes an interesting stuff so I decided to make a combination of both the blackboard and this tablet so I start with my first lecture is called vectors in plane and space you must probably already have learned something about geometrical vectors in school so something like this is a vector in space vector is planing something which I don't have to tell you but let me first start motivating this whole idea of functions of more than one variables and doing calculus on them you know one of the first thing that you learn in school addition so you are given two numbers and your arcs to add so put it in a more abstract way what are you doing you are just doing the following so I give you a number X and I give you a number I and there is a kind of machine calls an adding machine so let me write down this machine is an adding machine adding machine and then if you give me them input X and if you give me the input Y I give you an output Z so these are the inputs I give you two numbers and you add them and you give me a single number at the input so which I write as Z is equal to X plus y so when X say when X is equal to three and y equal to five Z it implies that Z is equal to eight the output is eight so this Z equal to X plus y can be written as f of X Y function of x and y I want to also mention that this notion of vectors is intimately linked with a function of more than one variables so you observe that here when I am adding so you are doing something in school so before you learn function of one variable you have essentially started learning functions of two variables and this is the one we really don't realize it and what I want to say is that why the notion of a vector is associated with this function of more than one variables for us let us just hold on to two variables because you see when you are writing a function of the form Z equals to X plus y you have two input variables basically what you give me you give me an input variable x and y this is my input variable and out comes the output so there is some operation here addition operation and outcomes the output Z which is X plus y so the input variable is not one real number but two real numbers so what lesson I learn here the first step is that input variable when I add two numbers input variable I can view it like this it consists of two real numbers right it just consists of two real numbers okay great now how does how do we represent two real numbers we represent them as a point on the plane as a Cartesian coordinate that is what you have learned in your basic school so here I draw it the drawings drawn become clear my straight lines are usually curvy lines as if my hand is shaking which sometimes does I can't just pull them straight for some strange reason so this is the x-axis which you know very Oh and this is the y-axis the vertical axis with this point being the origin 0 0 so any point X Y it can be anywhere I'm just putting it here in the first quadrant doesn't mean that is only in the first quadrant so any point XY I can view it as a geometrical vector just draw a line segment connecting 0 0 with XY and at the point XY put a direction so this XY so if I call this point as P which has coordinate XY this XY represents the vector o P XY represents the vector o P so the tip of the vector is represented by two real numbers so I can on sometimes write because o is always 0 0 if I write this as oh sorry are not given o is the origin o is 0 0 so opie vector can always be written as X Y great I think it's good up to here so the plane is a collection the plane is a collection of these points this whole two-dimensional plane and we call all these collections we give a name we call it the r2 space so R is the space of real numbers on which we did calculus or one variables so it is just a line with the origin and the numbers written on the left and right the negative ones to the left and the positive ones to the right and now are two which you see is the whole plane the plane of the board actually it's kind of infinitely we don't we want to imagine it like that so if a lot of things in is happening in your imagination which you might find very strange when you want to actually related with physical objects now that's likely a different story we will not get into that at this moment go so let me just write down that a 2-dimensional Cartesian plane is given as a collection of vectors of this form [Music] where X is in R and Y is in R now you might ask me why are you calling this Opie a vector so what is a vector you have learned in physics that vector is a quantity which has which objects an object not a quantity really an object which has both magnitude and direction so let me write down so it's the quantity having magnitude means a length length means length from origin magnitude and direction so let me just give an example look at this quantity o P here so Opie is in one direction suppose I can draw a vector of the same length as o P but we'll have a different direction so this can be portrayed by drawing a circle so give me some time to draw a circle sell draw a circle with center 0 and radius 1 so here I'll have x-axis so physics tells me vector or quanta objects which has both magnitude plus direction now hear this oh this vector o P has a magnitude which is the length of this vector o P which will come we'll come to later on in the next lecture how do we calculate length and all those stuff but it also other direction now now I'll show you that there are two objects two vectors which can have the same length but different direction and hence they're different vectors so two different points on a plane actually represents two different vectors so that's what's the key idea behind it so let me draw a circle like this so the circle which is given by the equation X s square plus y square equal to one the circle which has be centered at O the origin and has a radius one my drawing is very bad many of the viewers has much better drawing than me so I am sure that you would forgive me for my stupid drawings but I am trying to actually convey some mathematical idea so this is the circular boundary the circle so this is 1 0 this is minus 1 0 this is 0 minus 1 and this point is 0 1 now take 2 different points on the circumference of the circle set consider P which is X hat Y hat and Q which is X dash y dash now observe Opie and OQ so what is so you join with Oh both Oh piano Q so Opie is the radius of the circle so it's length is one because this has radius one OQ is also radius of the circle and hence has the length one because this is a circle of radius one that is what this equation says length one but look at the direction of Oh piano cue you can immediately see that the directions are different and so they cannot really present the same vector so that is why a point or an input for function of two variables is represented by a vector now how can I go and represent vectors in three dimensions of course you know that up to three dimensions everything is I can visualize and beyond three dimensions I may not be able to visualize of course you can't visualize then geometry becomes algebra you can just do many geometric manipulations our view to algebraic manipulations so when you are in three dimensions when you're in two dimensions you are talking about length and breadth when you are in three dimensions you are talking about length breadth and height so here I have three axis this is my x axis y axis and z axis so this is x axis y axis and z axis so any point you know three dimensional plane a three-dimensional space has three components length breadth and height and has to be represented by three numbers so here is any point P P is a kind of prototype point and this is having three numbers X Y Z so if you drop a perpendicular on the XY plane this XY plane and then from there it hits the XY plane at some point say P - and from P - you draw a perpendicular on the x axis and then you draw a perpendicular on the y axis then this length from the origin o to say Q this is Q dash and Q Double Dash then Q - oq o Q has a length x oq - as a length Y and P P - this one has a length z this one this this whole thing has a length Z so that is your length breadth and height so once you have created a model or with three variables so you can construct this is called r3 which is called the three dimensional Cartesian space so let me write down three dimensional now you might ask me what is the dimension is there a proper to your definition of a dimension yes but here you take on first our intuitive knowledge or intuitive view how we know what is the meaning of being in three dimension we ourselves are three dimensional objects sitting in a three dimensional space of this room length breadth and height so there is a three dimensional Cartesian space God Kardashian this name is coming after the famous French mathematician philosopher Rene Descartes a who invented coordinate geometry so how do I represent this any point is a triplet of numbers X Y Z where X is elemental or Y is element of R and Z is element of R now can there be a four dimensional space of course the modern theory of special relativity depends on four-dimensional space so special relativity for example Liuzza's what are called four-vectors so we're the first vector is time and the rest are the space vectors so that gives us makes the four dimensional space four dimensional system of space-time so special relativity where T is the time vector and X Y Z are the space vectors so the geometrical view of physical space is much better realized because time as Einstein showed that time is not absolute neither is space and everything is relative and hence it is much better visualize if we Club up time and space together but anyway we always are in our normal physical space is three-dimensional and when we are on time we get a four dimensional space so this it's natural to have spaces of more than three dimensions so in general mathematicians speak about an n-dimensional space and the model is called RN it is called the cut n dimensional Cartesian space so any vector in n dimensional Cartesian space wilco c11 sit in two dimension you have two two elements three dimension you have three elements and in n dimension you must have n elements so you have X 1 X 2 X n so that each X I is a real number where I is equal to 1 to n that's our model so such spaces are n which consists of vectors is RN so RN is often categorized as a kind of space which contains vectors and sometimes it is called a vector space vector space of dimension n now in geometrical vectors that you have learned how do you add two vectors you know when you when you do a geometry in two dimensions you know how do you add two vectors I'm just removing this part so you can easily have an idea so what do you do you add two vectors how do you add two vectors you knowing that in in the standard school high school vector geometry what you do is that okay you have two vectors one is say so this is your P this point is P with the coordinate X Y another is say so Oh P are there is Oh Q so given as say X dash y dash so how do you add it how do you add these two vectors so you draw a line from P parallel to OQ and so I draw a line parallel to OQ basically I'm from from Kew I draw a line parallel to Opie and there I from here I draw a line parallel talk you and they meet at a point say R so the new vector we joins oh and R is called the addition of the two vectors or a no Q so o R is viewed as an addition of Opie and OQ so we have Oh P plus Q so what is the coordinates which represents this vector if you are slightly if you take a little bit of time out of your busy schedule of today's life or you leave your mobile phone for quite a while you can immediately show that this coordinate is nothing but X plus X dashed will be the x coordinate and y plus y dash would be the y coordinate so any such space vector space of dimension n has two major operations that are important operations in RN so there are two operations in RN so what are those operations number one is called vector addition so how do I add two vectors you know ultimately the resultant vector would be represented by a vector whose ex corde x-coordinate will be the sum of the x coordinates and the y coordinate would be the sum of the y-coordinate the intuition that you have in two dimension would come up as I told you in the introductory lecture the difference essentially is between R and R two on once and the slight difference bernard and r3 but once you have some idea what are three you can have you have fair idea what are three the rest is algebra vector addition so what does vector addition mean so you have two vectors X x1 x2 xn in RN + y equal to Y 1 y 2 y n so these are the coordinates in RN and then the new resultant vector which is the addition of these two vectors has a coordinate so you add the first coordinate of X with the first coordinate of Y say it is x1 + y1 then the second coordinate is X 2 plus y 2 and then 8th coordinate is xn plus yn you see the same intuition what we did here we added the first coordinate with the first coordinate of P with the first coordinate of Q same coordinate of QP with the second coordinate of Q so you add the two x axis the first position and add the two Y positions the same thing we have done here you have just defined this whole thing then another operation which is called scalar multiplication basically you are you can now pull a vector or shrink a vector so here is a vector align you can pull it like a string and you can also shrink it down like a string so this called scalar multiplication scalar here for us would always be a real number so what is scalar multiplication what do we do with the scalar multiplication what we do is that you take an X take an X x1 x2 xn in RN and you take a lambda element of R then lambda times X is same as the vector lambda of X 1 lambda of X n so direction of this vector new vector lambda X is same as the direction of the vector X while the direction of the vector X plus y is like oh oh r has a different direction from both Oh piano q so X plus y has different directions from Z as a different direction from x and y but when you do a scalar multiplication lambda X has the same direction but the length only changes and this is also in RN okay so these are something which is typically absolutely so I have to also write for here you can understand that this is also in RN because this is the Topol of in real numbers so I without telling anything you should also able to know that this is n all right now these have some properties i I keep on telling all the properties it will bore you either you go to the text which is mentioned which I can show ya that this still available you can go to them and go to Amazon and buy it basic multivariable calculus it's an excellent excellent book I do not think there is anything which is better than this I have I haven't found any one which is written in a simple nice way for everyone to understand I guess the audience here is not just mathematicians it's all consists of physicists or chemists of engineers and mechanical engineers electrical engineers economists and statisticians and any maybe even from [Music] so I'm trying to make it look as much as simple as possible there but there are certain things about D with these properties you can do certain things one and for example so if you have an X as a vector X 1 X 2 X n now what I do I also I take a vector and I multiply 1 with X which I write as minus of X and this is the vector whose coordinates are now suppose I add these two vectors X plus minus X what do I get I get 0 0 0 it is a zero vector 0 vector which we can also write as a big 0 and another interesting fact is that if you now add 0 with any vector the addition is commutative either you write 0 + X or X plus 0 doesn't matter or X plus y and y plus X are the same this will make X remain as X so sometimes by taking a little advanced mathematical language borrowing it from group theory in algebra we call this as an additive identity so this is slightly more erudite language you may choose to use it you may not choose to use it and minus X by the way is called an additive inverse now what is more important is how can I represent any number say in three-dimensional space can I represent any not number any vector in three-dimensional space in terms of some basic vectors in terms of some standard simple vectors can I always do that and that's what I am going to show and that is what we are going to discuss which is called representation of vectors and I am going to use a notation which physicists would like and engineers will also like mathematicians who know some linear algebra might be by some luck might be disappointed but does not matter what I write is also can be read it very easily in the way modern mathematicians write representation of a vector so we will just bother about three dimensional vectors first I will leave you with a question of how do you represent n-dimensional vectors so in three dimensions which we already have there so here is a three dimension I again again I am drawing so there is little time which flows off as you draw it let me underline this so in this axis I will first draw three vectors so the point where I have the coordinate 1 0 0 I will call this vector I'll call this point C 1 and this is the origin and this point e 2 which will have 0 1 0 vector and the point e 3 on the z axis maybe not that we're going to four III which you will I will call it 0 0 0 0 1 vector so the three vectors Oh II owe you one Oh II - Oh III represents the point 1 0 0 0 1 0 and 0 0 1 respectively there if these vectors are representing those coordinates now this vector we 1 given by 1 0 0 is called the I've Ector by physicists so this is what physicists would call I've Ector and we 2 0 1 0 is called a J vector by a physicist and we 3 1 sorry 0 0 1 is called the K vector by physicists now I come back to the drawing board and let me draw so I now have the x axis the y axis and the z axis now let us consider any number P so in any I not number I'm so sorry any point P which is represented by XYZ the 3 coordinates so I connect them and complete the vector o P where o is the origin with 0 0 0 so the origin is always 0 0 0 I am just to remind you I'm writing it on the side so that you don't forget when you re see the video and here you have the AI vector here you have the J vector area of the K vector physicists would always like to call opieop osition vector the position vector of a point that is a vector o p is talking about is giving you the position of the point P in space so o P as many physicists would call as a position vector position vector of P now many physicists would like to write this position vector as R vector Oh P vector is equal to R vector how do I represent our vector now if you go by simple mathematics so now if you look at it very carefully I can write this vector X Y Z in the following way so X is a real number take X as a real number multiply it with 1 0 0 of the scalar and multiply scalar multiply Y with the vector 0 1 0 and scalar multiply the vector 0 0 1 with Zed and add them up that would give you this vector X Y Zed you can just try it out it is very simple now once you have done it you know that this is representing the R vector so this R is being represented by this coordinate X Y Z and this is I director this J vector this K vector so you can write what physicists always write that R vector is X into y vector Y into J vector and Z into K vector and this way of writing has a lot of advantage especially when doing applications in Natural Sciences and I'm an engineering and that's what we will really use in many many our applications we are not going to go beyond this today so just to end it I'm giving you a example now consider how do I represent a line for example in 3 dimension is there any way I can represent a line vectorially and write down its equation in three dimension you see how we can do it using vectors so take a line L and which passes through a point Q I am NOT a know is the origin here I am NOT drawing the access which has coordinates X naught y naught Z naught and another is P which is a call the running coordinates any any in any point P which is lying on this line L so from o you draw this position vector of Q OQ and the position vector of P o P now called see that this vector PQ and let us consider D vector and in this direction of L this parallel to L there is a vector D consider a vector D D vector which is parallel to L and so it DS is essentially along the same direction of PQ so D is say given as some real number T times P Q vector or P Q can be written as T into D right I you you'll soon realize that okay I have basically over some taken a vector D and if I want a lit up scale that or I can take a vector like this is a bigger one and I can scale it down or scale it up I will get a it could be in this direction D I can scale it in the negative way putting a negative number and I can get a make it exactly equal to PQ basically what you do you take the length of D you take the length of PQ you take the ratio and see what is T DQ by D should be T now if you look at the vector P you will immediately know from your basic geometry vector geometry that oq+ PQ is exactly equal to o P vector now what is oq o Q is represented by the coordinates X naught Y naught and Z naught and P Q is T into D and O P is XYZ now T now suppose D is the position vector of a point whose coordinates are a B and C so a B and C are basically lying on a line parallel to the line L then what would happen I can write this equation now becomes X not Y not Z naught plus T times a B C is equal to x y&z and zero so what do you have you have that X is nothing but X naught plus 80 y is nothing but y naught plus 80 and Z is nothing but Z naught plus 80 so this is exactly the equation of a straight line lying in the space lying in three-dimensional space in terms of the parameter T it is called the parametric representation of a straight line so what we have learned now is parametric representation of a straight line presentation of a straight and I am using shortcut in this what's up world shortcut is much more important so friends I think you have enjoyed this talk I hope so it's you'll decide and we I hope this mixture of the board and this makes much more sense and so let's end it up here today and the next lecture would be on lengths of vectors and inner products dot products ok thank you very much and good night [Music] [Music] [Music]