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Show that if $ riangle$ satisfies the 1 Lagrange equations, it identically satisfies $E_{4}$. (7.161) on the basis of the homogeneity of $A$. by explicitly forming...

Question

Show that if $ riangle$ satisfies the 1 Lagrange equations, it identically satisfies $E_{4}$. (7.161) on the basis of the homogeneity of $A$. by explicitly forming the total derivative with respect to $ heta$ that occurs in the equation.

Show that if $\triangle$ satisfies the 1 Lagrange equations, it identically satisfies $E_{4}$. (7.161) on the basis of the homogeneity of $A$. by explicitly forming the total derivative with respect to $\theta$ that occurs in the equation.



Answers

Prove each identity, assuming that $ S $and $ E $ satisfy the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second-order partial derivatives.

$ \displaystyle V(E) = \frac{1}{3}\iint_S \textbf{F} \cdot d\textbf{S} $,

where $ \textbf{F}(x, y, z) = x \, \textbf{i} + y \, \textbf{j} + z \, \textbf{k} $

So far, the Shinto we write out the right now the integral by divergent Siri Costis and and this bit of former divergence off F times. Constant factor is a service, Grady in off F tattle council doctor, which should be easy to verify. And, uh, because our Caesar consul that her we can point out and leave us with a new identity here.

Level. Oh, so for this problem, we're asked to show that this function ffx why has a relative maximum even though the discriminative is zero at that point. So we're looking for extreme points. So we start by doing our first derivatives. This is minus for X cube. That's the derivative respect to X. We do the derivative respect. Why, this is minus for why cute Now, critical points happen at points a B that make these two expressions zero. And it's fairly clear to see that the solution that we have is X comma Y is equal to 00 for only critical point. But in X equals zero y equals zero. Both these expressions are zero, and, uh and that's a critical point. So now what we want to do is want to calculate our discriminative. At that point, it's indiscriminate. Still find this way where the function is evaluated at our point 00 So to calculate these, we do more derivatives. So we do to drive. It's a suspect X. So what we're gonna have is we're gonna have a minus 12 x squared for too derivative respect. Why? We're gonna have a minus 12. Why square. This lets a good symmetry here and for this derivative right here. We're taking the Druid of of efs of X with respect to why this is a minus for X Cube, there's no wise there. So this is actually a zero. And you can see that when we evaluate these two expressions at X and Y equals zero, they're both equal to zero. So it was our discriminates. Eagle to zero. Now, normally, this would say that Okay, are indiscriminate Didn't tell us anything, so we can't conclude if it's a maximum The saddle point, our minimum. And so the way that we argue that we actually know there is a relative maximum at the 0.0 is if we look at the expression, we have a positive one and we have a subtraction. We subtract x of the forearm you subtract. Why, to the fore or these two terms since four is an even exponents The used to terms are always positive. Put a negative two for X you get for you but in two for X to get was sorry negative to you get 16 buzzard of 16. So this actually is one always minus something that's positive. So this is always going to be less than one. So we're always subtracting away from one there for all values of X and Y. So that means that the way to actually maximize this is this term is zero, which happens at zero comma, zero are critical point. And we know that if you change either of these numbers, if you make, you know, xnegative one, whatever it is, whatever you put into these this X and why you're always gonna make this function smaller and so, therefore are zero common zero is actually a maximum.

Okay, so we're told here that Ah, and step for everyone to form the system or equations and bags of Dex. Why? And the equal of Syria. And then Yeah. Why next? Why I am the zero and then f landa is X. Why Landa? Okay, So then what is that? Ex wife Linda, once the objective function is the function of trying to maximize or minimize minus landa times thie constrain equation that you want to be zero. Okay, so what do we get when we take FX? Well, we get f X minus Lambda gx. You want that to be zero? And then if y is f by minus Lambda Gee, why you want that to be zero. And as Linda, what do we get? We get f wait, actually, just get, um minus Jean. We want that to be zero. And so if you look at this, we get We need that. FX needs to be lumberjacks. F y I needs to be landed g y And then, of course, genes to be Theo


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