Transcripts
1. Introduction: welcome ladies and sentiment and scores. We're going to learn a more multi variable calculus, which is just the calculus off functions off. Several made you. But I'm your instructor, attaches to Misha, and I'm going to teach you the scores. France grass in an into doobie. At the end of this course, you will be able to solve the most difficult problems off multi variable calculus. Like a proof the topic said I'm gonna cover in the schools are partial differentiation, multi valuable team drool. Legrand's is multiplier, multiple integration Jacoby in and applications are multi variable. Calculus in this course have included lots off video electors to explain each and every concept supported by last number of examples. And at the end, off each section, I have included tests and quizzes along with solution so that you get most out off these lectures. This course is best for engineering, math, physics, major university students. And this course can also act as a quick refresher for professionals who seek knowledge in this subject. Thank you and see you in the course
2. Introduction to Multivariable Calcuus: When we talk about calculus, we usually mean calculus of single variable calculus that deals with functions off only one . We'd be able the distance travelled by a ball rolling on Krypton. This surface is only dependent upon we. It's a function off single, reliable. Where's multi variable calculus deals with functions involving several variables two or more than two relievers, for example, off X one x two and so on. Accent F off X Y is also a multi valuable function. Let's represent the distance. Seven in the first case, but the one off we an example off multi variable function pension. Off to reliable is the ball projected at an angle kita at an initial velocity. The distance traveled by this wall within a given interval of time depends Bhutan velocity and angle projection. It's a multi valuable function. The characterised that deals with function of two or more valuable is called market calculus.
3. Partial Differentiation: hello to do. We're going to learn about partial differentiation and has the route when we partially free in chilled off function FX one affects to where affects whenever. Next to our variables. With respect to X G, we consider all of the variables is constant Head. We have a function affects Why question X squared plus y squared. And when we partially differentiated with respect to X, we consider why this constant ex tone difference years normally but y squared becomes, see do because we consider it as constant. Similarly, when we partial differentiated with respect to why the ex tone becomes constant, so it's differentiation instead of rated becomes Edu and the White Downs different. Sheer It's not me.
4. Partial Differentiation-Example: Waas sub Ladies and gentlemen, Now we'll consider an example off partial differentiation. So here we will write f off X y equals two x Q by Les Y X This X square. Now we will. Partial difference. She ate dysfunction that respect to Why now? Ex called Autumn involving X will behaviors constant so excusable remain. Excuse that's real remaining and her ex square will become CEO because there's no white on now when we partially differentiated with respect to x ex cubic become three x square. Why will behave as constant X will become one and X square would become two X Now if we different shared this expression this result in expression again, with respect to why we'll get three x Square, it would be here like constant. Why will become one and two x will behave his conscience So it's debated will become zero I hope you clear with by sir difference change
5. Multivariable Chain Rule: we learn about multi variable change that's considered a function therefore, X way where X and Y are functions off d then partially innovative off effort Respect to t is equal to are so Did he wait you off? F with respect to X multiplied by parcel delivered you off x with respect to D plus partially innovative off f with respect to why much of glide by partially video off. Why, with respect to t first analyzed this for a French in off and redoubles where x one x two x in RBD bills, then partially weight you off f with respect to de is similarly equal to partially weight you off half wit respect to x one multiplied by partially way to off X one with respect ity place partial deliberate you off effort respect to x two multiplied by a partially we do off x two with respect to T and similar domes Do X And now let's take a look at an example which will further clear this concept. Let f off x y equal to ex quit this way X is a function of t which is t square less duty and why off t is equal to treat it Now. We'll find the radio partially radio off effort, respect, duty using our formula, the alerts, firing the individual partially weight do. And let's astute them in this formula, the start with partially redo off effort Respect to X, which is two X and we'll find similar tones. Possibly redo off X with respect to t. No, it's his duty and similar terms for why Effort? Respect away. This is one y with respect to t much is three and now we'll substitute all this partially did radio into our formula in order to find partial derivative off f with respect to D with her Stoute each and every ready and this gives has our end result. This is how we find partially redo off f with respect to t.
6. Multivariable Chain Rule-Example: Welcome back, ladies intended mint. Now I'll consider an example which will show multi variable change all its full glory So partially radio off f with respect to t is equal to possibly wait you off f with respect to X multiplied by partially radio off packs with respect to t plus partial the radio off effort respect to why multiplied by partial deliberative off. Why redress back to t Now I have a function which has accent Why as reliable and accent Why in turn depend upon de their functions off d So first of all, find partially radio off asks with respect to t Then I will find partially radio off. Why with respect to key which will give me three d squared and then I will find partially radio off f with respect to X it will give me too. It's then I'll find partially radio off effort respect to why? Which will give me one. Now I'll plug each and every value into the respective places to find partially weighty will half with respect to t using the elbow formula. So first I will take two X I'll multiply too partially read your acts with respect duty, which is to deep less too. Then I'll add this. Do one multiplied by partially radio. Why would trust back to T Registry D Square Now? I can easily simplify this and find the result in expression, which is the answer.
7. Method #1 to find maximum and minimum of a function-Elimination: I will end the first method of finding maximum minimum, which is elimination Has the problem find the minimum distance from the car y equals two minus X squared to the region. First, I'll represent distance by D, which is equal to rule over off X squared plus y square distance from the origin. And now f is quite of d was is this X squared plus y square and it is a function which is much easier to work with on it makes absolutely no difference whether we minimize d r f now eliminate why from the expression off f using the equation the curve. I'll replace the equation the curve into the expression off f. Now I'm gonna differentiate F which is the first half for mine finding many one more maximum and I went decided to zero. Now I'm gonna solve this equation, which will give me some zeros. First, let me simplify it for the A skating. I'm getting zeros in farm off X equals zero and best minus root over three by two. Now I'm gonna find the second or two. Do you read you off French and F to check whether these points are maximum or minimum when I lag zero into the expression of second order delivery to when I replace X equals zero, I'm getting minus six, which is less than zero. That means zero is a point of maximum. So we're going to discard it because we need to find the minimum distance. So we need to find them points off minimum. Now I'm gonna replace points plus minus route over three by two. Let me replace this in the expression. It's giving me three, which is obviously greater than zero within. Plus, these air points are minimum. Now I'm gonna find corresponding value off. Why? For the points plus minus root over three. It's one upon to. So our minimum distance is Rudy went off one upon to square my route over a three by two square, which gives us rude or seven by two. That's the answer
8. Method #2-Implicit Differentiation: Welcome back. Now we'll consider the matatu for finding many moment maximum off function, which is called implicit differentiation. So here we have a problem. Find the shortest distance between the curb. Why equals tu minus X squared and origin. So first relied expression for the distance which is rude over X squared plus y squared weekends have stewed this expression with F which is square off deep. So is it becomes a lot easier to work with this function rather than working directly with D was really differentiate f So we get two x t x Andi Do I d way now we divide throughout by DX so we get daily radio off after it Respect to X, which equals two x place Do I multiplied by D Y by d. Here. This is not in this city step, but we do it for the sake of clarity. Now we different shoot. Why the orders? Elko. So we get minus two x dx. We consider these two e creation on research suit equation number two into equation number one and place. Why reserve Stoute and please off d y re substitute minus two x dx. Now we again divide throughout. Now we again deride throughout by D x, which gives us an expression for daily radio off with respect to X. And now, in this expression, hadiya, my DX we serve shoot. Why, with the equation off additional cope and equated to zero in order to find minimum, we equated the deliberative to zero. And here we get an equation and we'll try to find the roots off this equation. The roots are let me call them. CDU's zeros are X equals Tzeitel and plus minus root over three by two. Now we have our expression for deliberative off F with respect to X, let me write it. And now, in order to find whether the zeros off this Wenshan are, why so many one more maximum, We will find the second order the video off F with respect to X. Now that's SEC. If the Zito's give us positive numbers on negative numbers, when we plug in X equals zero, which is our first serial off the equation, we get minus four, which is less than zero. So X equals zero is our point of maximum from basic calculus Course. We know that if the second already radio is less than zero. We get a point of maximum. Now we can plug plus minus root over three by two together because the expression for second or innovative in walls only X credit terms. So it makes no difference where they re dick, positive or negative number her own. Simplifying the expression we get minus expressed Will X Square. And when we substitute Akwa with plus minus root over three by two, we get nine and result in numbers three, which is greater than zero. So the point plus minus root over a three by two are the point of minimum. So in order to find the minimum distance but is rude over X square, plus why square we need to find what is why corresponding value. Why or X equals plus minus root over three. So we ride the expression for why we serve. Shoot X square by three by two and we get cough now researched. Shoot the values for X Square and Y square in this expression, and we get rude over off three by two plus half, which is equal to
9. Implicit Differentiation-Example: Hey, now consider this example when the shortest distance from the origin to the plain X minus y place that equals two. Now we will write the immigration or distance from the origin with his square Rudolph X squared plus y Square Aziz. Great. Now we can also right half which is equal to the square equals X Square plus y square inserts great. Instead, off minimizing d, we can instead minimize f which will make our work easier and it makes absolutely no difference. And now we will serve Stoute the value of acts from the equation Pain in two F in please off X square. Well, it's have Stoute the expression for acts from the equation of plane and plug it into X Square. And now we will partial different sheet f with respect to y said corporate, consider now we'll set it equal to zero in orderto minimize this punch. Simplifying this we get hopeless Four way man ist ooze it. This is our equation one now would you are so do prints You'd f we trust back to said which gives us minus four ones. Do y bless who has said this is our equation too These are our two summered in this equation, which saw one and two and find the values for why and said it gives us. Sadiq was two y three and like was minus two up on three. Now substituting Z and why into equation on the plane, we get X Who's one? My tree? Our minimum distance. Now these are minimum. It's clear that these values air for minimum because we're getting only unique values. Andi, the distance is least so we won't go into the deal's off. Whether checking whether these are minimum or maximum, we'll find directly find the minimum square to doff, which gives us one unit, which is our ends.
10. Method #3 -Lagrange Multipliers: welcome Mary Wim. This is the last month had it is called a grungy multiplayer is the best matter. So usually to find the minimum and maximum were given two things crushed French. An FX right with was many more maximum value. This to be deter might and on expression, sigh X y equals constant, which is relation between X and y. It is also called constraint. So first we'll find the daily radio off fi in the terms of different Shiels. This is just the usual process for finding the, um the ready wolf a function while the real you with puncheon and no and it is equal to zero because it's equal to constant and the rebate you off a constant is always equal to zero. And now we will find the declarative off F D f and again right the usual expression and good against Said this to Zito. Why? Because we want to minimize dysfunction Now we will multiply. The first equation would Lambda Lambda as a multi player and it gave the manta its name and now we'll do an interesting thing we will add would off these equation the right hand sides of the aggressions. After multiplying Lambda, we learned the writer in side of the equation, which will give us an interesting result. In the first expression, we have partially waiting with respect to X. And in the second, under the second bracket, we'll have the parcel. Way to respect why, nor the pattern here in the under the first bracket we have partial. Do we do with respect to X and another second record, we have partial deteriorated with respect to why Inwood kisses the partial innovative off Why, with respect to X or why is multiplied by multiplayer Lambda. Now we will set boot off these expression and a bracket equal to zero because, uh, in the right hand side the d X tone and D y done our CTO. So we said them equal to see toe so parts of the radio off with respect to X. Thus them that times Parsons re video of five with respect to X equals zero and we have our second equation. We get this equation because in the right and tired off the expression, there's no DX 30 way, so we need to set them in equal to zero. That's the need. I hope you understand that I'm getting bit regress here. But don't worry. Everything will be clear in a matter of minutes. And now all right, the relation between X and Y And had we get three creation 123 we will solve for X Y and Lambda and thes X Y are actually the points off minimum and maximum for the function. We really don't need Lambda, But in order to find X Y sometimes we need to find Lander to now we'll consider Ah, very easy way to do this entire thing without getting involved in all this Relight a function f off X y equals Air Force X y bless Lambda Times, fire off X Y and we find the partial derivative off upper case F with respect to X partially innovative off f with respect to why and we have our relation between X and right , this is our relation between accident. You saw all this and find X y and Lambda and trust me guys, this very easy mattered. Once you'll see the problem will understand it fully
11. Lagrange Multipliers-Example 1: Welcome back. No will consider a familiar example off the previous section, which is finding distance between go y equals two minus X square on the origin to illustrate the mattered off flag ranges multiply. We'll verify whether we're getting the same points or not, so that's first right. The function f off X y equals X squared plus y squared. It is the square of the distance between the curve and origin. We have choose this because it's easier to work with. And now we will try to find for your fax cycles conscience, which is the relation between X and y. With nothing but the equation of our crew, we will put the constant on the right hand side and the relation between X and Y on the left hand side. This is the right way, and now I will show you the wrong way to do this. This is putting the equation as it is now. I will express a function f X y as the summer off, Small F X Way and Lambda Times Fi. Now it's astute f off X way with X squared plus y squared and fire ex way with Y plus X squared have we will not right toe. Remember that because that's the constant part. Will let this will only date the relation between X and Y. Now we'll parcel differentiated with respect to X, which will give me an equation. And I was said to 20 then a bombshell differentiated with respect to why and I will against at 20 now, from the first equation, you find that X is equal to zero, and Lambda equals minus form. Atma righted. Now if we said shoot Lambda equals minus one into the second equation, we get two lamb, two y equals one which gives me why equals half. Now if I substitute, why calls half into the equation? Off Go. What I get is X equals plus minus three bite. These are exactly the same point which we have often in our previous problem. So this sues that lack grandes multiplier works very well
12. Lagrange Multipliers-Example 2: Hello. Everyone now would consider another example on lag ranches multiplier. So ask a star. Find the volume the largest box with as is battling to the axis inscribed in the spear X squared plus y squared plus z squared equals R Square. Her. I have the figure with the box inscribed inside this beer with the edges, Berlin to the access and the land off each spear is two x. Try to that, respectively, and the reason I'm using to her because the center of the spear and the books are going sitting. So the warnings to works much blind by two y multiply y to Z, which is eight x y Z. Now I will write the function half off. X y Z equals F plus lam no dimes fly her f is the warning off the box, which is eight x y z and FYI is X squared plus y squared crazy square. We will ignore our square hair because while taking five, we ignore the constant. So I'll put the values off F on the respective value off. I and I will barter difference. You'd f with respect to X Ali created 20 Then I will parcel different shoot f with respect to why And I would again equated to zero. And finally I find the partial delivered you off. After what respect is it equal Asiedu? Now what I'm gonna do is I would multiply the first equation with acts The second equation with why, on the last equation would see I will name the equations 123 Now, by my to playing this, what I'm gonna get is and the first equation I'll get eight x y z Bless to Luanda X Q. Similarly, in the second equation, I will again get eight x y Z plus lambda into two y square On In the tart equation, I will again get eight x y Z plus lambda into two Z square. Now, if I add each of these equation, what I'm gonna get is three into X y said, because because all the x y z Dems off all three equation and together and then to Lambda X squared plus y square plus C square. All the terms accumulator together equals zero. Now X squared plus y squared plus Z squared is equal to our square around the equation. This pure in that X squared plus y squared plus z squared is equal to our square To Harry, right to longer times r squared this from degree shot This beer Now I will find value off Lambda using the sick wishin with his minus two Will x Y z upon are square Now I will write the equation number one Do we remember the equation? Number one? I will again right it right I did. I'll do this to find the value off X. Now put the value of Lambda into this equation which will give me a twic minus 24 X square Y Z up one are square equals zero. Now on solving this equation on getting X squared equals one by three r squared. Now this is the maximum value off X that we can get. That's clear. We can't get excreted in this. So there is no point on checking whether these are points off. Many more, more maximum and enemies were getting only one value because distance can't be negative. Similarly, on solving each of the equation, we get y square cost one upon three are square and see square coast one upon three years square on using the same procedure as in this one which gives us X equals. Why cause equals one upon Route three are we have taken positive here because distance can't be negative. So maximum waddling is equal to age X y Z, which is eight multiplied by three route tree, one by tree to tree into our to this is the maximum William off the box inside is beautiful .
13. Multiple Integration-Introduction: Welcome back. We will try to develop an institution for multiple in TV, so let's start with normal definite in deep plants in normal definite indeed girls. In order to find the media under co, we divide the car into small strips off bread delta acts and land affects. So the area of this rectangle is land tam bright, which is FX times del. Tax. Now the total area is equal to the sum off all smaller areas, which is equal to somewhere F o X I dimes. Doug attacks. This is like finding the yeah, Richie's black Nitties slice Rickert the teas lies into smaller pieces and then we some all those area to find media off the entities slice. Now, when Delta extends to zero, the submission sign turns into integrations and and dull tax becomes detox. No, in case off multiple integral reconsider cylinder And we tried to find the volume off this cylinder in order to dollars into Shin. So how we have a function which defines this cylinder which is Z off X y and Blanding. Miss Land Times area, which is Z off x y dance Delta E. Now Doug Day is de extends D Y, which is the idea Off cross section off a small box you can see out there there's a small box was lenders large, but the cross sexually area is very small. Now the entire volume is some off all smaller values. We use double submission here because post three, some over one access then we somewhere another access which gives us Z off X y times Delta X Delta Way Now, in case off integration, we write the submission Signers integration sign Andre, multiply dx dy y. Now the land off this small box which I have got through in this volume is the ex wife and the cross sectional media. He's DX town d y. So Delta V is Z off X y dance dx. By the way. Now this element off why exes is D y and another element in the XX is is DX and had this is how we get the wall. You off this
14. Multiple Integration-Example 1: her will consider another example on multiple integration. Find the walling bound by the plane through Express y plus Z equals pool, lying to express why calls to on the court any taxes her have drawn the figure in order to find the warding off the figure, I will divide volume into small slabs off volume Delta V. Delta Bi is the wall you want to slap when an ad each of the slabs one each of Islam, I get the volume off the entire figure. So Delta V, which is the one on the slam, is Edie at times thickness. Now what is the area to slap? This is integration off after 19 degrees in off, said Danced. Ey. Why? Because this is parallel duda z y plane. I'll put the limit from why Zito Do Tu minus two works because the equation the Line two X plus y, equals two. I get to minus two X. Why, where is from? Zero. Do why and wise to my honest works times that thickness, which is Delta X now, is some the volume off each slab, which is some mission off Delta bi putting the limits. Why time still dags now the submission signs Jeans is too integration, saying When Delta X approaches e had I have considered acts to be constant and why is reading? We can do it other way around too. Axes rating from 0 to 1, you can see from the base figure. The last figure is the bees off the figure or is not a fever Axes waiting from 0 to 1. Now what is said? We did and said from the equation off a plane, let me call it See we derives from the equation of the plane. Now we will do the double integration off this expression. Her Why is not constant because we have considered X constant here and why is wearing from 0 to 2 months to works, which is the equation on the line? Always remember that we put the value off. See here now in double integration when we integrate with respect to why we will consider access constant so I can take four minus two works is constant and directly multiply. But why? Why will become Why square by two, which is the usual integration process Now I'll put the limits. Integration with respect to acts will remain Stilgar. It won't go anywhere now. I'll substitute the limits when we integrate with respect to variables we consider all of the really well, says Constant. This process is quite similar to partial difference in equation partial differentiation. Whenever you integrate with respect to available, consider all of the available is constant. Now we have substituted liquids. Now we'll simply fight the reason why limits off acts are constant because the slab was barrel the when and the slab stick nous was delta attacks. And there he had considered X to be constant. And why we're going from 0 to 2, minus two works now. We will integrate this with respect to X This expression with respect to X After simplification, we will degrade this expression with respect to X, which gives her six sex minus four X square. This do what excuse Continue Macedo to one. No,
15. Limits of Integration: using the last problem will now learn how to decide the limits off integration. Here we have the base off the figure off The last problem It is in the why x X is here. We have constructed same line Why plus two X equals two with the slab off thickness. Delta eggs had the limits for X rays from 0 to 1. That's very clear because endpoint for X here as we move ahead, the endpoint for X is one. But here in this lab, we really don't know where does why end in the slab? Extra means constant, but why is variable? But we don't know the end point of why So we right Why equals two minus two X using the equation the line. Similarly, as we move ahead along the one you we still don't know what is the end point off white along that slab. So again we consider why equals two minus two X, but still X is constant and the limit varies from 0 to 1. We can do other way around and do the same thing for why an ex reversed, have we no extra means constant. So why remains constant and X varies from zero to a non point. So we do not the limit off X by one minus. Why were two using the same increase in plain? Who are why the limit ranges from 0 to 2 hair? Also, as we move ahead along the volume, we still have no idea what is the end point off X. So we again right limit zero from from 0 to 1 minus y by two. And that's how we were.
16. Multiple Integration-Example 2: Now we will see one more example. Here we have X and we need integrated with respect to y and X. First, we will integrate it with respect to y because in the firstengy grin, the integration limits are off. Why? From zero to route over one minus X square so fast we need to integrate it with respect. Why we will consider access Constant, which will give us X y. Now we will substitute the limits. The limits off. Why that really give us X multiplied by Ludewig off one minus X square DX Now this is normal. Definite integral. We will substitute one month's X square by T Square and two x minus two X dx equals two to DDT. It will give us X dx minus cause minus DDT. Now we'll substitute X dx with my nasty DT and one minus X square by square. And the limits will be tinged. We will. So Stoute No, This is the normal tickle we saw usually give us take you by three Modesty Humor tree on substituting the limits will get one by three
17. Multiple Integration-Example 3: Hey, Now, as one more example on multiple anti grills, find the wall you includes for the parable I'd see equal 16 months X square minus y squared and the points or 00 a 12 and be 10 Now draw the B softies three dimensional figure. The base is located at the X Y axis. I draw the blinds school A and B, respectively. And here we have a triangle. We have a triangular base now equation off the line. You We'll try to find the equation off line you because we're going to need it in our future calculations. Using the intercept formula, you find the equation of the line you Which is why Quills two X now. Well, imagine that we're cutting a slab through the figure through it. Three damaged and figure. And when we add this lamps, we cover the entire warning. The limits for why I will be wearing from zero to Twix. Because why is were you able here and access constant from the equation of the line y equals two X. We have limit from zero to do. X X varies from 0 to 1. Now we will expand Z in its full form and the first kiss will consider acts as constant. And we will integrate with respect to why. And that's why we have taken the limits from 0 to 2 works. Because why is really able here? Since why is the first integral here will first integrate with respect to why considering access Constant, I was put the limits putting the limits. We get 32 X minus four X cubed minus age X cube by three. Now we'll integrate it with respect to X. This is just normal integration. West will simply fight and now we will integrate. It gives us 16 x square minus five upon three X power for lemons from 0 to 1. In the end, we get what, 23 upon three Unit que. This is the wall You want the figure bond by parable light
18. Jacobians: come back in this lecture. We're gonna talk about Jakub. Ian's now considered the situation. Let's say that in two dimensions. Accent. Why are functions off new evils be in Cuba than the Jacoby in off X Y, With respect to peak you is, Jacoby in is usually represented by letter G. It is the determinant off a matrix In the first rule, the partial differentiate the variable X with respect to new variables. B and Q. And similarly, in the second rule, the partial differentiate. The really able why would respect to new radio, but B and Q. Now, whenever there is a change in variables like we consider change of variables from one coordinate system to another co ordinate system, we ride the area element the new area element as Jakub Ian dinz the differentials off the new variables that is Jacoby in times deep ridicule. Similarly, in Three Dimension, where X, Y and Z are presented as movie Riebel, B Q and R, the Jacoby in is a determined off three by three matrix. Since in the earlier case, we had considered two by two matrix because there there were only two real evils. But here we have three variables, so we will consider three by three matrix in the first. True, be partial difference here with respect to new variables in each room. Each old radial this partial differentiated with respect toe each of the new real levels. What should we have X second, we have why and third truly have said. Now here we find the new volume element that is Devi using this Jacoby in, we can find the new Radia Bill the new volume element. It's J times, db de que and DEA, which is the product off differentials off new value. So Jacoby and becomes very useful whenever there is change of Real's.
19. Jacobians-Example: a hair. We'll see an example on Jacoby in when there's a Q B in off x y with prospective polar coordinates. R anteed up. So we will begin with writing Accent. Why, as functions off our and Tita X is our cost theater. And why is our science, Tita? Now we'll try to find Jakub Ian off X y with respect to our and teeter You're right. The formula off the Jacoby in man tricks with his DX by d r. Then again DX by D. Tita In the first room, we differentiate acts with respect to the new variables. And in the second room, we differentiate with why would respect to the new real humans. Now we will replace with the values will replace the value off acts and why which will give us when we differentiate in the first column with respect to our our gets eliminated in the second column The difference yet with respect to Thida. So the trigonometry crunch in guards altered. Now when you find the determinant of this matrix, we see that it's are Cost Square. Tita plus sci cortina now cost square teed up lessons. Cortina is equal to one soared Jacoby in is are now the Edie Element is Jacoby. In times the differentials off new malleable. So it is our d a r D Tita.
20. Application of Multivariable Calculus: Chelsea An application off multi, Really? With calculus. So Alaska started from the moment of inertia Off right. Circular cone with height s and then city room. Harold, consider cylindrical coordinates the start off by finding moss off this body. It is density dance volume. Here we will consider Z teacher and our, which is cylindrical coordinates. Z will vary from zero to etch which is the height off the coun and our is going to vary from zero to the y Z because at each and every point the read years off this corn is equal to its height and T days reading from 0 to 2 pi in order to cover the entire circumference off the coun and Harry right RDR, Dita and DZ Why are because our is the Jacoby in off X y Z with respect to cylindrical coordinates? That's why we write our here. You can really fight yourself. And now we have already sold for Tita and are and we're going to solve for Z, which gives us a rope. I ask you upon three her by X cube is warning on tycoon and rules the density productive to gives us the mass off this body. Now we'll find moment of inertia with respect to Z axis, Moment of inertia as mass times. Distance square And it's somewhere. So here we will use multiple, integral, specifically triple integral the limits. Who would be essentially the ship? The seam? Her We're using our square because we're considering the distance, which is our square moment of inertia is mass. Times did stand square. That's why we have considered are square here, an artist, the Jacoby in Now we have again already tiled for Tita and are and I'm going to solve for the This gives us Z to depart five by 20 multiplied by by turns to buy. Now we'll serve Shoot the limits which gives us rule by edge to the par five divided by 10 . Now we will substitute the value of rule from the moss equation. At first we have found out the mosque. Now we will right rue intense a mass and we will substitute in this moment of inertia. Rule is tm by that cube and solving this it gives us three upon 10 am a square. This is the moment of inertia off a corn