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Consider F and C below.F(x, y, z)= yz i + xz j +(xy + 4z) kC is the line segment from (3, 0, −2) to(4, 4, 3)(a) Find a function f suchthat F = ∇f.f(x, y...

Question

Consider F and C below.F(x, y, z)= yz i + xz j +(xy + 4z) kC is the line segment from (3, 0, −2) to(4, 4, 3)(a) Find a function f suchthat F = ∇f.f(x, y, z) =(b) Use part (a) to evaluate C∇f · dr along the givencurve C.

Consider F and C below. F(x, y, z) = yz i + xz j + (xy + 4z) k C is the line segment from (3, 0, −2) to (4, 4, 3) (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.



Answers

(a) Find a function $ f $ such that $ \textbf{F} = \nabla f $ and (b) use part (a) to evaluate $ \int_C \textbf{F} \cdot d \textbf{r} $ along the given curve $ C $.

$ \textbf{F}(x, y, z) = yz \, \textbf{i} + xz \, \textbf{j} + (xy + 2z) \textbf{k} $,
$ C $ is the line segment from $ (1, 0, -2) $ to $ (4, 6, 3) $

If this problem were given the short information our 1st 7 this might be to make sure that our vector field f is conservative And to make sure this everyone has said this component hearing which is in Foshan peeing this component equals some function Que and this component here iniquitous, um, function on and we're gonna make sure that the partial of P with respect to why is equal to the partial of Q respect tags. We're also going to make sure that the partial of P with respect to see is equal to the partial of our respect to X. And last, they were going to make sure the partial of Q with respect to Z is equal to the partial of our with respect to y another partial of playing with respect a wise going to be hey, partial of Q with respect to eggs is also Z. So that is good. A partial of p with respect, disease going to be why partial of our with respect to X is also going to be why so that was good. The partial of cure with respect to zem is going to be X the partial of our with their sector. Why is also going to be X So therefore the vector field is conservative now we confined our function nets. And to do this we are going to put our component p in an integral and evaluated with DX. So Y Z the X I don't mean way want to get but its function in why in the woods with a parcel of this in terms of us, that derivative of G of some wide of the emcee would be zero. And now we are going to takes a partial of this. Why? When we do this, we get X Z plus Jeep Crime y Z, and we're going to set that equal to our function Que which is xz. Therefore, we can see the G prime of y Z is equal to zero. And if you put that an integral with zero de y, we're going to get some constant, which in this case, is actually going to be some of function in Z in a Z on Lee. So God is going to be a function in Z Onley, which is conservator constant. And so I'm going to call the H of Z and so up here. I'm going to write that in and we get X Z cause Age of Z. Now I'm going to take the a partial of this in terms of Z. And when I do guns, excuse me, I'm going to actually not plug it in right there. Plug it in over here. And so I'm actually going to get X y z plus hz because G of y Z is equal to agency. And so again we get X y Z plus h of C. I'm going to set that equal to our But before I do that I first of the take partial of it in terms of Z. And so hey, the partial of this in terms of Z and I get X y. Thus it crime of C. And now this is when I said it equal to our just x y plus Tuesday. Therefore, we can see the age primacy secrets Tuesday. And if we put Susan and cover you see, we go z squared because some constancy and this is equal to H A Z, which we can put in appear and we see that our function of some X y Z is equal to X y Z plus h of Z, which again we stand out is Z squared host, see? And so that is our answer to part A Now for part B, or we're going to do this. Cigarettes are endpoint for 63 So good in to what we just got So 463 against Herbie. We're looking for 63 We get four times, six times three plus three squared. We don't need to include the constant because we're going to subtract what we get when we put in the next point. That's gonna just erase the constant. And so we get four times six is 24 times three of 72 post nine is 81 and then we're gonna plug in the next point, which is 10 negative, too. And when we close this in, we get one tons zero times negative to that zero plus negative two squared. That's four. And so we get forward. And now we're going to subtract four from 81 and we're going to get 77. And that is our answer to party

Okay. So we had the um vector field F. Of X. Supply Z. Is equal to the following Y. Z. Hi hat plus X. C. J. Hat plus why? I'm sorry X. Y. Shit plus two Z. Can. Yeah. In this case we know that um were given that F. Is this conservative gradient field. So then um we also given that we want to solve the senate role over the path of a line segment. Mhm. Going from 10 native to two 463 Yeah. And so what we can do is we are given this great. It's just great fields. We know these these uh three derivatives. We know the F over D. X. Is equal to Y. Z. DF or Dubai question. I have a question why is equal to at seeing and then partial F. Partial Z. Is X. Y plus two Z. So we can integrate you two of these two to compare. So we have from this X equation we have the F is equal to X, Y. Z. Plus some function G. Of Y. Z. From the second equation we have physical X. Y. Z plus some function H backs nZ. And then from this last equation we have actually go to X. Y. Z, pussy squared. And then let's say some function. Uh huh. Jay. Of why? I'm sorry X. Y. And so we'll notice comparing all these equations, they all share the same X. Y. Z. Part and these two are missing disease. Great part. So that means that G. Of Y. Z. Is equal to H. Of X. C. And in this case that is just equal to she's cleared since we're not worried about any constant terms. So in this case we have this F. Of X. Y. Sorry I faced Y. Z. Is just equal to X. Y. Z, pussy squared. And so we can solve this integral. We already said it's going from this point. Uh see is this um 10 -2? All two way too 463 Uh huh. So we can just go ahead and use that. We know that this is a gradient field. Yes sugar. We could just sell this as the f of 463 minus half of one's your negative too. Which in this case gives us four times 6. I'm three plus three squared minus one Time zero times native to um mine. Sorry Plus -2 Squared. So we have 24 Times three. Um giving us 72 plus three squared is nine-. We have negative 00 plus floor and so some minus and equals. Mhm. So we have 72 plus nine minus four Which just gives us um 77 as the answer to this integral.

Were given a vector field Big F and A Curve C where, as defined in part a function little f such the big F is equal to the ingredients of little F. So big f is the function y Z I plus x zj plus X y plus two zk and C is the line segment from 10 negative too. 2463 Now in part A. We know right away that the partial derivative of Little Left with respect to X is equal to the X component of big F, which is Y Z. And this implies that little F of X y Z is equal to x times y times z plus some function G of y and Z, and therefore it follows that the partial derivative of little Left with respect toe why this is equal to X Z, plus the partial derivative of G. With respect to why. But we also know that this should be equal to the why components of big F, which is xz. Therefore we have that the partial derivative of G, with respect to why is equal to zero, and this implies that function G is equal to some function h of z to take the anti derivative with respect toe. Why therefore something this together we have that little less is equal to X y Z plus h NC and taking the derivative with respect Dizzy little left with respect, dizzy is equal to X y plus h prime of Z. This is also equal to the Z component of big F, which is X y plus two Z. Therefore, it follows that H prime of Z is equal to two Z or in other words, HFC is equal to Z squared, plus some constant K. Now, if we take a to be equal to zero, we have that little left is equal to X y Z plus Z square in part B, whereas to evaluate the line integral of big F along the curve C by the fundamental the're, um for line into girls. The line integral of big F along the curve C is equal to little less at the terminal in 0.463 minus little f at the initial endpoint 10 negative, too. Plugging these into our above formula For little left, we get 81 minus four, which is 77

Yeah. In this problem we want to find the first partial reruns of the function out of x. Y. Z is x squared Y cute plus two X Y Z minus three Y Z at the point negative +212 This question is challenging understanding of differentiation of multi varied functions. To solve you understand that partial readers require us to use single variable differentiation techniques where we differentiate within something differentiating variable and which we all other variables and constants. So with this in mind we can proceed to solve the example given now in red showcase what we need to do in terms of functions and maps perfectly onto the power rule and constant rule for the function F X, Y Z. That's why the power role we have X equals two X. Like you plus two, I c minus zero. That's why is three X squared Y squared plus two X z minus three Z and Z is zero plus two xy minus three Y. The zero term occurs when you differentiate with respect to a variable that's not present in the term for fx at zero where there's no X. For +00 where there's no Z. That's plugging into negative 212 We have F X negative 212 is negative four times one plus two times one times two minus zero equals zero. F Y is three times four times one plus two times negative two times two minus three times two is negative two and f z is zero plus negative four minus seven equals negative seven.


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