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2) (20) pts ) Use partial derivatives to decide whether O not the vector field F(x,y) (6x?e"+xe2r +2xeY _ 3e ) conservative. (Show necessary calculations bef...

Question

2) (20) pts ) Use partial derivatives to decide whether O not the vector field F(x,y) (6x?e"+xe2r +2xeY _ 3e ) conservative. (Show necessary calculations before giving your conclusion:)

2) (20) pts ) Use partial derivatives to decide whether O not the vector field F(x,y) (6x?e"+xe2r +2xeY _ 3e ) conservative. (Show necessary calculations before giving your conclusion:)



Answers

$3-10$ Determine whether or not $\mathbf{F}$ is a conservative vector field.
If it is, find a function $f$ such that $\mathbf{F}=\nabla f .$
$$\mathbf{F}(x, y)=\left(x y+y^{2}\right) \mathbf{i}+\left(x^{2}+2 x y\right) \mathbf{j}$$

Okay, so we're looking at the vector field F f x Y is equal to why squared minus two X. I had plus two X. Y. Yeah. J hash You want to see if this is a conservative vector field? So so we can do that pipe checking to see if the second derivatives of equal, because if this was a vector field that was a gradient field and this would be of the form uh huh. X derivative of X F times I had plus the wide review of that time she had. So if we go ahead and find The 2nd derivative which uh F X Y. The second derivative F I X after equal. This means that this is in fact a great field that we can solve for. Let's go ahead and do that. So we have um you can find that's sorry the partial with respect to Y. Yeah, of Y squared minus two X. And see if that's equal to the partial perspective tax of two x. Y. We'll see that. This is just equal to why and that this is also equal to Y. So that yes, this is a conservative vector field. Yeah. So basically now what we can do is we can take uh we can write this of the form if it's just a crazy an event. So like I said earlier this is just a derivative. Mhm. So we know that um DF over dx is just equal to um Y squared minus two X. Yeah, This is why I spared -2 x. And that partial F. Partial line is just equal to X. Y. So if we integrate both sides we'll see that um this side leaves us with um partial F is equal to y squared minus two X. Partial X. And then we can integrate to be left with F is equal to x. Y squared minus two Xy plus some uh G function of why? Because if we take the partial respect to X. Uh this is our integration constant. Our integration integration term. Okay so that's our first equation. If you do the second equation we'll see that partial F. Is equal to two X. Y. Sorry to Yeah two x. y. Partial. Why leaving us with integration? And so F. Is equal to um X. Y squared plus some H function will say H. H. Of X. And so we wanted to see how these are equal. Yeah and so that we can see that um dysfunction in this age of X must be equal to -2 x. by yeah and then plus some constant. And we'll see that G. Of Y. It's just the constant. So they are. Yeah but that's why I just equal to um X. Y. Sorry xy squared uh minus two Xy But um Constancy. Oh

Were given a vector field Big F, and were asked to determine whether or not Big F is a conservative vector field if it is, whereas to find a scaler function. Little left such that big F is three Grady int of little F. The Vector Field Big F is natural Log of y plus two x y Cube I plus three x squared y squared plus X divided by Why J. Well, we have the partial derivative of the X component natural log of Y plus two x y Cube. With respect to why is one over why plus six X y squared and the partial derivative of the Y component three X squared y squared plus X divided by why partial derivative with respect to X is also equal to one over X plus six x y squared. Notice that the vector Field F has a domain which is set of all payers x y such that why must be strictly greater than zero. This is because of the natural log of why the X component. This is a half plane which we know is open and simply connected. Therefore, by a the're, um from this section it follows that the vector field Big F is conservative and by another the're, um it follows that big AF is equal to I'm sorry. There exists a scaler function little left such that big f is equal to the Grady int of little Left. This equation implies that the partial derivative of little Left with respect to X is equal to the X component of Big F Natural Log. Why plus three X I guess it's two x y cube. Sorry. And that the partial derivative little left with respect to why is equal to the y component of big F, which is three X squared y squared plus X over y for the left equation. If we take the anti derivative with respect to X, we have that f of X y is equal to x times the natural log of why plus X squared. Why cubed plus some function g of why and taking the partial derivative with respect toe why we get X over y plus three x squared. Why squared plus g prime of y tweeting this to our other partial derivative of little left with respect to why we get three x squared, why squared plus X over y solving they get that g prime of y is equal to zero and taking the anti derivative G f y is some constant putting this all together We have that a potential function. Little laugh is equal to x times the natural log of y plus X cubed. Why squared plus K. I'm sorry. That should be X squared. Why cute? Yeah, This is a potential function for big F.

Were given a vector field. Big eth were asked to determine whether or not Big F is a conservative vector field, if it is, were asked to find a function little left such that big F is the radiant of little left. So the vector field big F is two x y plus. Why, to the negative second I plus X squared minus two X wide of the negative third J and why is always greater than zero? We have the partial derivative of the X component to X y, plus wider the negative. Second, with respect to why is two X minus two y to the negative third, and we have that the partial derivative of the Y component X squared minus two x Y. The negative third partial derivative of this with respect to X is likewise equal to X minus two times why to the negative third, and we have the domain of our vector field. Big F. This is the set of all pairs x y such that why is greater than zero. So essentially it's the half plane, which, because why is strictly great in zero is open, and that's a half plane is simply connected therefore, by a serum from this section we have that are vector field Big F is conservative. And so by another theory, um, we have that there exists a scaler function little left such that big f I'm sorry. Big F is equal to the radiant of little less Now this implies that the partial derivative of little left with respect to X must be equal to the X component a big F, which is two x y plus wire, the negative second and that the partial derivative of little left with respect to why must be equal to the y component Big F, which is X squared minus two Ex wife The negative Third, Taking the anti derivative of the equation on the left With respect to X, we have that f f x y is equal to x squared Why plus x y to the negative second, plus some function g of why and therefore taking the derivative with respect to why we have that The partial derivative of little left with respect toe why is equal to x squared minus two times x times y to the negative third plus g prime of y comparing us to other expression for the partial derivative of F. Little left with respect to why this is equal to X squared minus two x times. Why to the negative third. And this implies that G prime of y is equal to zero, or that g of y is equal to the constant K. Putting this all together we have that little less is equal to X squared. Why, plus x y to the negative. Second plus K. This is a potential function for big guests.

Okay, so you want to go ahead analyze this vector field big alphabets and Y. Is equal to L. N. Y plus Y. Over X. Times I had plus L. N. X plus X over Y. J. I. And you can see there is some symmetry with X and Y. Um, across the two components. So this gives us a clue that this might be conservative. But we have we have to actually check. So if this conservative, we know if we call this just p times I had plus Q. Times she had we know that F is conservative. Yeah. If partial of P respect to Y is equal to the partial key respect to acts. Yeah. And um this also implies that F. Is this gradient field of this little funk dysfunction their mouth? So let's go ahead and take those derivatives. We can do partial P. Or partial. Why? It's just partial partial Why of natural log of why plus Y over X. And we know that we have this Ellen. Why are you giving us one over Y and Y over X. Is why is just a con why is the um immigration. I mean the differentiating variable, we know that uh one of Brexit is constant. So it's just a linear term leaving us with one over X. And then next we want to do partial peculiar partial X. Who just partial partial X. M. L. N. X. X. Or Y. Which gives us a similar results leaving us with um one over X. Plus one of my. And so these are clearly equal. So that sends deep a partial partial wise, you get a partial Q partial X. That means F is conservative. Okay. And that F is this gradient field of little F. So then we want to find this gradient field uh find this uh, function from the gradient field. So, we know that if F. Is this gradient field, that means that um the components of that have to equal a partial uhh F partial X. Times I had times my plus partial life over partial Why times J. Hob. And so now we have this equality and we know that these derivatives we know partial left in the partial X. You just see what Q. And that Q is just natural, log Y plus Y. Or X. And then we know partial F partial Y. He's also sorry, this is not gonna be cute. Is he going to peak? So P is the is the I had component Q. Is the jihad component, This is natural log X plus X over Y. So here we have these two qualities. We know that um we can go ahead and separate these and integrate. So we know that we have partial life is just L. N, Y Plus Y over X. Partial X. And so if we integrate both. Higher spec tactics we're gonna be left with have physical to this case X. L. N. Y. Is constant. So we have X. L. N. Y. And then Y over X. Um gives us uh why times one over X. So we have why time is going to go over one direct which is natural large objects. And then we have to add this integration constant. This integration term H. Of Y. His H function which would go away if we were to it if we took the derivative respect to X. And then we also want to do the wild component. So we have partial F. Is equal to natural long X. X over Y. Partial. Why let me in the rain Both sides leaving us with physical too. And the same thing as before. We have Y. L N. X plus exhale and why? Plus this function achieved alecks. So now we can see that the results from both derivatives give us the same exact values about that with these added constants or functions of eight X and Y. But we can see that since all of our functions of X and Y are contained here, there's no possible way that there can be both. That put uh this G and H functions have any dependence on the X and Y. Because I would um ruined the that would mess up the symmetry of the F functions over the derivatives. So we're just left with G F X, Z, quota H of Y. And we know that this is both just a constant term that's not depending on either X or Y. So finally we know that this function F. Of X. Y. Of which this big factor field, destructive big is a gradient field of is equal to Y l n X plus X, l n Y plus some integration. Constancy, Yeah.


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