5

53-54 Verify that the conclusion of Clairaut's Theorem holds. that is. Uxy Uyx _ 53. x sin(x + 2y) 54. = xty? 2xys...

Question

53-54 Verify that the conclusion of Clairaut's Theorem holds. that is. Uxy Uyx _ 53. x sin(x + 2y) 54. = xty? 2xys

53-54 Verify that the conclusion of Clairaut's Theorem holds. that is. Uxy Uyx _ 53. x sin(x + 2y) 54. = xty? 2xys



Answers

Verify that the conclusion of Clairaut's Theorem holds, that is, $ u_{xy} = u_{yx} $.
$ u = \cos (x^2y) $

Alright, So to begin, let's just note that the derivative of CoSine is negative. Sign. Let's begin by calculating partial u partial X, which is going to be, um, minus two x sine x squared. Why? And now that's calculate the why partial of that which is going to be. And that's note that, um, the derivative, uh, sign is going to be co sign. Alright, cool. Now let's calculate the y derivative of that which is basically going to give us, um minus two x cubed cosine X squared. Why? Now, let's calculate the why partial of you, which is going to give us minus X squared. Uh, sine x squared. Why? And now we can calculate you y x, which is going to give us negative to X cubed cosign, X squared. Why? Now we see that these air equal and we have verified Claros, the're, um

For this problem, we are asked to verify that the conclusion of Claros theorem holds that is that the second partial or the mixed partial derivative is independent of order. So first we want to take the partial derivative with respect to X. We'll need to apply the product rule. So derivative of X is just going to be one. So we have sine of X plus two Y. Then we multiply X by the derivative of sine of X plus two Y. Derivative of Sine is going to be positive coast. Uh So we'll have X. Coasts of X plus two Y. We would multiply by the derivative of the inside of sign of expose to why. But that is just one. Then we take the partial derivative with respect to why? Where we'll only need to apply chain rule here. The derivative of the inside is just going to be too. So we get to X. Sine of X plus two Y. Now taking the derivative of taking the derivative with respect to X. And then with respect to Y. We get the derivative of sine of X plus two. Y is going to be two Sine of X plus two Y. And then the derivative of X times coast of X plus two Y. It's going to be plus or actually it's going to be end up being -2. X. coast of X plus two. Y. Excuse me not coast. Sine of X plus two Y. Then taking the derivative with respect to Why? First, then with respect to X we'll need to apply the product rule. So we first have to sign of X plus two Y. From taking the derivative with respect to X. And multiplying it by to sign X plus two Y. And then we need to add on the derivative of uh two X. Sine of X plus two Y. With respect to X. One second here. I thought something seemed off the partial, the first partial with respect to why should have been coast there. So we should have here too. Then let's see, okay, need deposit. And this guy should have been a coast as well. Okay, now all seems right. So most are taking the derivative of why or with respect to Y. And then with respect to X gives us to coast of X plus two Y. And then we need to subtract off because or rather we'll be adding on technically two X times the derivative of coasts of X plus two Y. With respect to X. That is going to be negative sine of X plus two Y times one. So we end up with minus two X. Sine of X plus two Y. And we can see that the two mixed parcels are equal

The function we look here is next to the first Y. Three minus Y. Z. In game. So we are going to prove X. Y. Z is equal to U. X. Y. Is equal to U. Y. X. So this is equivalent to prove that. Yeah. So the pressure differential asian partial X. And the partial Y. Is equal to pressure you over pressure. Why pressure is So So we first compute partial you partial X. So that's his Speaker two. So because this term contains X. So this change didn't sell for suggesting it is for X cube. Why cute? And this is zero into if we compute how sure you in the past away. So the first hand has a Y. And the second team has a wife. So for the first time. So that is three X. Four wide square minus four. Why cute? So here we go to do another partial differentiation. That is partial partial wine for this partial you pressure X. So when we take the pressure differential for why? So this turn only contents wine into that will be four X. Cubed Times three White Square. So this will vitriol Xq widescreen. And for here we're going to take some partial differentiation for X. So that is. And this one. So we see this term contains expertise. This train didn't and for this time it is three time floor X cube time widescreen. And this will be chow xq white square. And we see the tear tends their echo. So that means that this set aside in this. Were you satisfied

To verify that the conclusion of the theorem holds for you equal to erased. X. Y times sine of Y. We first want to find the first partial derivatives of you in terms of X. And in terms of Y. So for the first partial derivative of you. In terms of X. We have partial derivative in terms of X. Of erased Xy times sine of Y. In this case sign of wise constant since Y is a constant variable. And so we have a sign of why this is multiply too erased. X. Y. Times why. And that's the same as why. Sign of wine Times erase two X. Y. And then for the first partial derivative of you. In terms of why we have partial derivative in terms of Y. Of the race to X. Y times sine of white. And in here we have to apply a product rule and we have E. Res two X. Y times the derivative of sine. Y. That's co sign of Y. Plus we have sign of Y times the derivative of erased X. Y. That will be erased. Xy times the derivative of X. Y. Which is X. And so we have the race to X. Y times co sign of Y plus X. Sign of Y. Now, if it take the partial derivative in terms of Y. E. For the first partial derivative of you. In terms of X. We have partial derivative in terms of why of why sign of white times erased X. Y. And here we apply product rule and we have why sign of why times derivative of erase Xy That's erase X. Y. Times X. And then from here we have plus erased Xy. This times derivative of why sign a boy. We apply again product rule and that's why co sign of why plus side of Y. And And here we have erased two X. Y. Times we have X. Y. Sign Y plus. Why co sign of wine plus sign of why Now for when you take the partial derivative in terms of X. Are the first partial derivative of you. In terms of why we have partial derivative. In terms of X. Of erase the X. Y. Times go sign of Y plus X. Sign of Y. This is just erased. Xy times zero Plus sign of Y. Plus we have co sign of wine plus X. Sign of Y. These times E. Race to X. Y times Y. And we have E. Res two X. Y. This times sine of Y plus why co sign of Y plus xy sign of Y. Now this is equal to This one. And so we have shown that the crm holds


Similar Solved Questions

5 answers
19- Write = the 2-ethyl-4-mcthyl-3 oxdation [ciclio oxohexanal, polassiuin and dichromale nalle organic - ptoduci20- Write - the esterification 2-ethylpentanoic reaction with sulfuric acid and acid catalyst for: propan-2-ol, and name the organc product.
19- Write = the 2-ethyl-4-mcthyl-3 oxdation [ciclio oxohexanal, polassiuin and dichromale nalle organic - ptoduci 20- Write - the esterification 2-ethylpentanoic reaction with sulfuric acid and acid catalyst for: propan-2-ol, and name the organc product....
5 answers
3 Solve the following systemn of Inequalitles by %raphine,2 > < 1-6 30/44>8-{-2 Glasa |
3 Solve the following systemn of Inequalitles by %raphine, 2 > < 1-6 3 0/44 >8-{-2 Glasa |...
4 answers
(08 pts.) Shown below is the 'H NMR spectrum fr compound with molecular formula C H,_Oz. The IR spectrum shows peaks at 1580 cm ' 1690 cm"' (very strung); vcry broad peak centered at 3000 cm and. hint of & peak at 3100 cm" Propose struclure.PPA
(08 pts.) Shown below is the 'H NMR spectrum fr compound with molecular formula C H,_Oz. The IR spectrum shows peaks at 1580 cm ' 1690 cm"' (very strung); vcry broad peak centered at 3000 cm and. hint of & peak at 3100 cm" Propose struclure. PPA...
5 answers
Tek R Ior 4e reg*cn bounded He Cafuq s 3 '4X 4 ^4 Cs 02 X24 Cio +R Uoluo 2 6 f S6qa 3ececr ( (qdslJ Y Dladud +0 (cne La Ler RAu0 Kon 4= 2V=Washr metlo (Use
Tek R Ior 4e reg*cn bounded He Cafuq s 3 '4X 4 ^4 Cs 02 X24 Cio +R Uoluo 2 6 f S6qa 3ececr ( (qdslJ Y Dladud +0 (cne La Ler RAu0 Kon 4= 2 V= Washr metlo ( Use...
5 answers
Which of the following is not a valid set of quantum numbers?n = 2,/=1,ml = 1,ms = -1/2n =4,/=0,ml = 1,ms =+1/2n = 5,/ =1,ml = 0,ms =+1/2n = 1,/= 0,ml = 0, ms = +1/2n=3,1=2, ml =-1,ms =-1/2
Which of the following is not a valid set of quantum numbers? n = 2,/=1,ml = 1,ms = -1/2 n =4,/=0,ml = 1,ms =+1/2 n = 5,/ =1,ml = 0,ms =+1/2 n = 1,/= 0,ml = 0, ms = +1/2 n=3,1=2, ml =-1,ms =-1/2...
5 answers
Mfa Question: 5 pts7 of 10 (8 complete)Find the area between the curves 13/12 Y#X y=IIx 1/12The area between the curves is (Do not round until the fnal answer Then round to the nearest thousandth as needed )
Mfa Question: 5 pts 7 of 10 (8 complete) Find the area between the curves 13/12 Y#X y=IIx 1/12 The area between the curves is (Do not round until the fnal answer Then round to the nearest thousandth as needed )...
4 answers
[NazSOa when 2 What J the deep lanced blue [ 'of Vanadium precipitate il 2 that forms? "Sloi 1 equation for the reaction_ (V(NO ) (aqlli: pappe solution 1 1(p) Write balanced net ionic equation for the reaction_Write ] ionic equation for the reaction
[NazSOa when 2 What J the deep lanced blue [ 'of Vanadium precipitate il 2 that forms? "Sloi 1 equation for the reaction_ (V(NO ) (aqlli: pappe solution 1 1 (p) Write balanced net ionic equation for the reaction_ Write ] ionic equation for the reaction...
5 answers
Apjda n,h;c cahitlcrixle tanmtanus ", !e; U: fonkcivon Tim Wnisky dcterminantini (Wronckian} he5Z4iZyz [Compute the Wronsky determinant (Wronskian) of the functions 91, 92, Y3 defined below with constants @,b,c]: Y1(2) = Ta + 81; y2(2) = rln(br); y:(r) = cr?Lutfen birini secin: 14ac In(ax)8bx 14ab In(bx)Zbx 14ab In(bx)14cX + 8ac In(cx)8cX 14ac In(bx)
Apjda n,h;c cahitlcrixle tanmtanus ", !e; U: fonkcivon Tim Wnisky dcterminantini (Wronckian} he5Z4iZyz [Compute the Wronsky determinant (Wronskian) of the functions 91, 92, Y3 defined below with constants @,b,c]: Y1(2) = Ta + 81; y2(2) = rln(br); y:(r) = cr? Lutfen birini secin: 14ac In(ax) 8bx...
5 answers
Aldehydes and ketones are distinguished by using(a) Tollen's reagent(b) Lucas reagent(c) Borshe reagent(d) all of these
Aldehydes and ketones are distinguished by using (a) Tollen's reagent (b) Lucas reagent (c) Borshe reagent (d) all of these...
5 answers
Write Lewis structures that obey the octet rule for each of the following, and assign oxidation numbers and formal charges to each atom: (a) $mathrm{NO}^{+}$, (b) $mathrm{POCl}_{3}$ (P is bonded to the three $mathrm{Cl}$ atoms and to the $mathrm{O}$ ), (c) $mathrm{ClO}_{4}^{-}$, (d) $mathrm{HClO}_{3}(mathrm{H}$ is bonded to $mathrm{O})$.
Write Lewis structures that obey the octet rule for each of the following, and assign oxidation numbers and formal charges to each atom: (a) $mathrm{NO}^{+}$, (b) $mathrm{POCl}_{3}$ (P is bonded to the three $mathrm{Cl}$ atoms and to the $mathrm{O}$ ), (c) $mathrm{ClO}_{4}^{-}$, (d) $mathrm{HClO}_{3...
5 answers
Evaluate the integral: (Use C for the constant of integration:)In(Vx) dx
Evaluate the integral: (Use C for the constant of integration:) In(Vx) dx...
1 answers
A point $P$ is moving along the curve whose equation is $y=\sqrt{x^{3}+17}$. When $P$ is at $(2,5), y$ is increasing at the rate of 2 units/s. How fast is $x$ changing?
A point $P$ is moving along the curve whose equation is $y=\sqrt{x^{3}+17}$. When $P$ is at $(2,5), y$ is increasing at the rate of 2 units/s. How fast is $x$ changing?...
5 answers
What are the main differences in the nature of the findingsderived from correlational and causational statistical tests? What are the four necessary preconditions for establishingcausality? ( 150 words) The answer must be based on Statistic forBusiness.
What are the main differences in the nature of the findings derived from correlational and causational statistical tests? What are the four necessary preconditions for establishing causality? ( 150 words) The answer must be based on Statistic for Business....
5 answers
Domain:Domain:Range:Range:Y Intercept:X Intercept:Horizontal Asymptote:Vertical Asymptote:
Domain: Domain: Range: Range: Y Intercept: X Intercept: Horizontal Asymptote: Vertical Asymptote:...
5 answers
3.) [46 points] Differentiate and fully simplify the following: a.) f(c) = =COS (2)Vzz 6.) g(x) =2+5 c) h(z) = tan (523) d.) f() = tan? (52) e.) y = arctan( V32) f.) j(y) = (In (y) + 5y) 42 9.) f(z) = aresin(sin (w) In (2)) h.) 2xy +6x + 4y3 42 make sure Yon solve explicitly for y' or d, whichever notation You preler: 9 sin (2)Vr 92 make sure YOu solve explicitly for Y and f'(x).
3.) [46 points] Differentiate and fully simplify the following: a.) f(c) = =COS (2)Vzz 6.) g(x) =2+5 c) h(z) = tan (523) d.) f() = tan? (52) e.) y = arctan( V32) f.) j(y) = (In (y) + 5y) 42 9.) f(z) = aresin(sin (w) In (2)) h.) 2xy +6x + 4y3 42 make sure Yon solve explicitly for y' or d, whiche...

-- 0.062377--