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11. This problem is all about vector fields. The three parts are independent_Determine without searching for a potential function) whether the following vec- tor fi...

Question

11. This problem is all about vector fields. The three parts are independent_Determine without searching for a potential function) whether the following vec- tor field is conservative. Your answer should be yes O no, followed by your (brief) reasoning: F = (z? yr ,y2 _ xy, 22 xy) (b) G = (2ry32' , 3x2y2 24 , 4x?y323 + 2) is a conservative vector field. Find the poten- tial function g(w,y, 2) such that 9(0,0,0) = 0. (c) h(x , Y,2) = 22 +y? + 22 is a potential function for conservative vecto

11. This problem is all about vector fields. The three parts are independent_ Determine without searching for a potential function) whether the following vec- tor field is conservative. Your answer should be yes O no, followed by your (brief) reasoning: F = (z? yr ,y2 _ xy, 22 xy) (b) G = (2ry32' , 3x2y2 24 , 4x?y323 + 2) is a conservative vector field. Find the poten- tial function g(w,y, 2) such that 9(0,0,0) = 0. (c) h(x , Y,2) = 22 +y? + 22 is a potential function for conservative vector field H = (2x,2y,22) . C is an unknown curve joining (0,0,0) to (1,2,3). Compute the work done in moving along C in the presence of H. 12 Consider the vector field F = (M,N,P) with M = y2? cos(zy) + In(2 + 1), N = rz? cos(ry) + 3 sin(2)eBysin(e) , P = 22 sin(ry) + + 3y cos( = 2)esy sin(e) 2 +1 Without finding the potential function; show that F is conservative. Find the potential function; 0, for the force. Evaluate J,1,o) (0,7/27) F dr



Answers

In Exercises $33-42,$ determine whether the vector field is conservative. If it is, find a potential function for the vector field. $$\mathbf{F}(x, y)=e^{x}(\cos y \mathbf{i}-\sin y \mathbf{j})$$

In this problem of vector field we have to determine whether the vector field F is conservative or not. And if it is conservative we have to find a potential function for the vector field and we have the vector field F of X, Y and Z as equals two. Why is squares eddy cue by plus two X. This is two X Y Z Q G Plus three XY Square. Save the script multiply with K. So for this first we have to find the value of Colonel F. And if Colonel F is equal to zero. So we say that the vector field is conservative. So this age I. G and K partial everybody with respect to X partially Ready with respect to Y and partially related with respect to that. Now coefficient effects that age there's a coefficient of I actually. So this is why square that is that cube coefficient of C which is two X Y, z q coefficient of K which is three X y squared square now value. So that's where they will be. I now we have to do the partial differentiation of three X y squared square with respect to Y. So which is six X Y. Is that the square minus now partially narrative of two X Y Z. Q. With respect to their So that's where they would be six. Excellent. Is that the square plus G. And this value of minus actually so this is minus G and multiplied with partially renovated all three X y squared ready square with respect to X. So this eight equals two three wise squares a square minus now partially reason of why squares NQ with respect to their So they say again three Y square zero square now plus key now partially with your two ex wives and you with respect to X is equal to this is no here to Y. Is that cube minus now person derivative of Y square that with respect to Y. So this is again two. Why is that you? So now this term is alone zero. So this team terms alone zero and this term is alone zero. So when we added some zero plus zero plus zero. So this value is again zero. So we have the value of curly hair. Physical zero. And if the value of Carlos Physical zero. So we say that the vector field is conservative and now so this age conservative vector field. Now we have to find the value put himself in some. So from here we see that F of X Y. Z are differentiated with respect to X. S equals two Y squared red cube F of X. Y. Zed differentiated with respect to Y is equal to coefficient of the which is two X. Y. Thank you. And F of X. Y Z are differentiated with respect to said is equal to three X Y squared square. Now we have to integrate any of the functions. So integration of severe taking X. So this really would be Y squared Z Q B X. So this really would be X Y squared Z Q plus of constant care. So we have the potential function as X Y squared Z Q plus Cape. So this is required put in cell phones and vector field it comes.

In this problem of vector field we have to determine whether the vector field is conservative or not. And if it is conservative we have to find a potential function for the vector field. So we have the given vector field F of X. Y. Is equal to X. Y plus Y. G divided with X squared plus Y square. And now we have to find the value of M. And M. Is the coefficient of ice. So this is X divided with x squared plus Y square. And we have to find the value of and and and and is the coefficient of the which is why divided with X squared plus Y square. And now we have to find the value of partial differentiation of em with respect to Y and partial differentiation of and with respect to X. And now when we perform the partial differentiation. So this term is equals two, her minus one divided with x squared plus y square. Holy square. And now this is X treated as constant. So this valuable reacts. And now we have to do the differentiation of wide square which is two ways. So this is multiplied with two Y. So we have minus two X Y divided with x squared plus y square. The holy square. Similarly we have to do the differentiation of and with respect to X. So that's where they would be hit X squared plus y squared divided with Holy Square. They said one divided -1 actually. So they said -1 divided with x squared plus y squared. Holy square. And now we have to treat why is constant. So this age why? And now we have to do the differentiation of extra square. So which is two weeks from here the value is minus two Xy divided with x squared plus Y is square. The whole square from here we say that the value of partial differentiation of em with respect to y is equal to partial differentiation of and with respect to X. So that's why we say that the function is conservative so we have the right answer is the answer is conservative. And now we have to find the value of putting self answer. So for this we have F of X Y. That means the differentiation of FX Way with respect to X is equals two. C X divided with x squared plus y square. And we have the value of say f of x Y differentiated with respect to Y is equal to why divided with x squared plus y squared. Now we have to integrate so when we integrate say F of X Y now integrated with respect to X. So this value will be integration X divided with x squared plus y squared. If here integrated with respect to X which is equal to. So we have to first we have to set up the differentiation of excess square in the numerator. So that's where they would be two weeks if we are multiplying tools so we have to divide it with two. So this where there were two X dx divided with X squared plus y square. Now this way it would be one divide with two and log off X squared plus Y squared plus a. Well, okay, similarly, we have to do with F off xy differentiated with respect to Y, and we will get the same value, so we have the value off potential function is one divide with two. L n X square plus. Why is square plus K? So this is the right answer where K is a constant.

In this problem of vector fields, we have to determine whether the vector field is conservative and if it is conservative we have to find a potential for himself for the vector field and we have the vector field, F f x y is equal to 15 white cube, i minus five XY Square G. And now from here we say that M is equal to 15 Y cube and n is equal to minus five X Y square. Now we have to find the value of partial differentiation of em with respect to Y and partial differentiation of N with respect to X. So when we do the differentiation, so that's where they would be 15, multiplied with three. So this way we will be 45 Y square. And when we do the differentiation of minus five X. Y squared with respect to X. So this way they would be minus five Y square. From here we say that partial differentiation of em with respect to Y is not equal to partial differentiation of end with respect to X. So that's why we say that this is not not conservative. Since this age, not conservative, there would be not any potential function. So this is not conservative. Yes.

In this problem of vector fields we have to determine whether the vector field is conservative and if it is conservative we have to find a potential function for the vector field. And we have directed field therefore of X. Y. Is equal so X into the power X square multiplied its way And whole multiply with two I plus XJ. So from here we say that M is equals two x. And they delivered with two xy multiplied with the to the power X square way and n. Is equal to visit access square into the power excess square way. And now we have to find the value of partial differentiation of em with respect to Y and partial differentiation of and with respect to X. No we have to do their partial differentiation. So this would be equal to we have to differentiate with respect to Y. So we have to provide we have to use here Product Rule. So this age two x. And this is two X definition of why is one and this value will lead to the power X squared Y plus. And now we have to keep this term as as it is. So this way they would be two X. Y. And differentiation of it is the power X squared Y is here into the power X squared away. And now the differentiation of X squared away is X squared. So from here we say that this is equals two X to the power X squared Y plus. To this valley will be executed. So this age X cube, why Italy power excess square white. And now when we do the differentiation of here partial definition of and with respect to X. So this way they will be against him. So this is two x. And into the power excess square wife bless again. This way there will be excess square. And here we have to do the differentiation of extra square Y. This is two X. Y. So this way would be to xy multiplied texas square is to execute by and Italy power X squared away from here we say that partial differentiation of em with respect to Y is equal to partial differentiation of end with respect to X. So we see that the vector field is conservative. And since this is conservative we have to find the value of Prince Alfonso. So from here we say that F. Of X way differentiated with respect to X is equal to hear this term is X. This is two x. y. E. To the power extra square. Right now we have to integrate it with respect to head. We have to integrate with respect to X. So integration of two X. Y. Z. To the power X squared away with respect to X is equals two. So when we differently when we integrated so this value comes to weigh E to the power X squared Y. And the constant constant or reconsider and the constant is also there say Kay so we have the potential fans and as E to the power X squared Y plus key. So this is the right answer yeah.


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