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Determine whether O not the vector field F (€,y,2) = In y; E + Inz. % is conservative. If it is conservative; find a function f such that F = Vf....

Question

Determine whether O not the vector field F (€,y,2) = In y; E + Inz. % is conservative. If it is conservative; find a function f such that F = Vf.

Determine whether O not the vector field F (€,y,2) = In y; E + Inz. % is conservative. If it is conservative; find a function f such that F = Vf.



Answers

Determine whether or not the vector field is conservative. If it is conservative, find a function $ f $ such that $ \textbf{F} = \nabla f $.

$ \textbf{F}(x, y, z) = e^{yz} \, \textbf{i} + xze^{yz} \, \textbf{j} + xye^{yz} \, \textbf{k} $

In this problem, you're given the showing information and were asked the first show if F is a conservative vector field. So the way that we're going to do this is to set this component here equal to some function p and this component here equal to some function que and we're gonna see if the partial of P, with respect to why is equal to the partial of Q. With respect to X. And when we do this, we see that the partial of P with respect to Y is going to be either the X Times co sign of why and then a partial of Q with respect to X is going to be e to the X co sign of why. So they are in effect equal. And so we know that, yes, it is a conservative vector field. And then to find the function F that they were looking for, we're going to do the following. We're going to take our function p and write an integral and then evaluate this in terms of X. So we're going to write DX. And when we saw this and take the anti derivative, we're going to get he did the X sign of why plus some function and why? Because if we took the partial of this with respect to X, the derivative of some g of why would be equal to zero, which is why we can to death. And then we're going to take it the partial of this new function with respect to why. And when we do this, we get E to the X co sign of why fourth g, prime of why and then what we're going to do with this is actually set it equal to our component for J or a function cute. And so when we do that, we get that eat of the X co sign of lie. Those g prime of why is equal to eat of the X Co sign off. Why this means the G prime of why is equal to zero Now we can use this to solve for G of lie by putting this into an integral and evaluating in terms of why and so we're going to get zero. Do you? Why? And that just means that our function g of why is equal to some constant. I'll call it K. And now I'm going to plug this back in to what I got up here. And when I do that, I see that our function f of sin X Y is equal to eat in the X No sign of why plus K which again I got my first taking the function p running in an integral with DX in a from the resulting anti derivative that took the partial their sector Why and then said that equal Teoh my function queue up here. I didn't use that to so for GPA fly. And then he respected the G of lie And then I plugged that back in. Do you have my final answer?

In this problem. We are given these shown information and were first asked to show if F is a conservative doctor field there. Where they were going to do this is to set this component equal to some function p and this component equal to some function que and to see if the partial of p, with respect to why is equal to the partial of Q with respect to X. And so when we take the partial of P with respect to why we're going to get e to the X Times negative sign of why and we'll see if that is equal to the partial of key with respect to X, which is going to be e to the X sign of why now we can see that these are not equal because this is equal to negative either the extent of why and this is critical to eat of the X sign of why so, therefore, they are not equal, and this is not a conservative vector field, and that is all we need or this problem

Determine whether or not this is a conservative vector Fields. So it takes off derivative off first term. With respect, sir, Why we should get this. I take that. The review was seven terms with respect to X, we shows okay, it was ex. So there's a SIM which means is it is a conservative after field and then we try to find the potential function or the function there, take the roots or take the greedy and becomes this factor field. We want partial f X because why it was X impartial f partial. Why he goes, He's with explosives with the white. This gives us f host entitle eruptive sorrow just itself. Constant can be any functional flight because we are taking the route through with respectful X. Similarly, this one it was axis treatise consuls or or we take entitle ripped It looks like this and tied the rope to is his self consul can be any function off. Why start any function of X pickles who are taking the route with respect to why on dh comparing this you we get f equals. Why eat with X and this one should be part of the Jew on the fly, So process it with a y and we always have any arbitrary, constant

So whether this is a conservative factor field, we checked the derivative of the first term with respect to why we have to use some product rule in general, the relative of E to the X Y should be X into the X y with respect to y and derivative of this term. Yeah, with respect to X Uh huh. So we should have y equals x y plus one plus x y times 30 of this with respect. Who? Why soul should be. Why so with respect to X should be y e to the X y So that's chip We should have to y e to the x y plus x y square into the XY So they are the same. So this is a conservative vector field. Then we have to figure out how this scalar function that takes gradient becomes this vector field. So we won't, uh, why screw it to the X y I plus one plus x y. It was XY Shea. That means the relative with respect to X by the definition of gradient is this derivative with respect to why, by the definition of the gradient should be this. So we just have to find the, uh, find anti derivative. Well, f should be why it was X y. Because when you take the derivative, you multiply by another wine plus and constant and counsel in this case should be should be any function of why he also obtained the relative with respect to only X. How about this one? We want to find the the derivative with respect to, um with respect to, uh why was this? So we have to find anti the report with respect to Why, of course, you can use a trick like integration by parents, etcetera. But I think a good guess is you start by because they are going to match at the end. We might as well just guess it will have some form like this. Then we try to see if they match. We take the derivative with respect to why we got X. Y e e e to the x y plus x y uh huh. Plus XY start into the XY plus you're multiplying xing from X Y E to the X Y byproduct role, which is exactly this six and constant could be any function of X. I mention these two. We know these who are just constant function so we can conclude that f equals y. It was the X Y, plus a constant.


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