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Evaluate the line integraldr by evaluating the surface integral in Stokes' Theorem with an appropriate choice of $.Assume that has counterclockwise orientation...

Question

Evaluate the line integraldr by evaluating the surface integral in Stokes' Theorem with an appropriate choice of $.Assume that has counterclockwise orientation when viewed from above(6ry = sin 2,3x? sin z 3X7y cosz/ ; C is the boundary of the plane z = 6 _ 2x - Jy in the first octantfF-dr-0 (Type an exact answer )

Evaluate the line integral dr by evaluating the surface integral in Stokes' Theorem with an appropriate choice of $.Assume that has counterclockwise orientation when viewed from above (6ry = sin 2,3x? sin z 3X7y cosz/ ; C is the boundary of the plane z = 6 _ 2x - Jy in the first octant fF-dr-0 (Type an exact answer )



Answers

Evaluate the line integral $\oint_{C} \mathbf{F} \cdot$ dr by evaluating the surface integral in Stokes" Theorem with an appropriate choice of S. Assume C has a counterclockwise orientation. $\mathbf{F}=\left\langle 2 x y \sin z, x^{2} \sin z, x^{2} y \cos z\right\rangle ; C$ is the boundary of the plane $z=8-2 x-4 y$ in the first octant.

Problem. Number 16. Let's see the the closed curve C a story, the surfaces closes and F vector field Stokes Theorem says that the line on the surface integral are equal. In other words, we can say that the by an integral off F D X equals the surface off the curl of the vector Field F and yes, to evaluate the line integral, it is enough to evaluate the surface integral. It requires computing the Girl of the Vector Field F She is equal to i, G and K partial derivative with respect to X portion derivative with respect why and fortune derivative with respect to sent two X Y signs, said X squared. Sign That and X squared. Why co signs that peace, which is equal toe the partial derivative off X squared? Why cool signs it with respect. Oh, why minus the partner and derivative of X squared signs that with respect to that, the portion derivative off two x y sign that with respect to Zed minus the partial derivative of X squared, why co signs that with respect Toe X partial derivative of X square sign, then with respect to X minus partial derivative to X Y signs that with respect Oh boy, this is equal to X squared co sign Z minus X squared course Shinzi two x y who signed Z minus two x y Good signs you into X sign the minus two x Why fine. Zini, To get to find an answer off, you know it wins you. The surface integral equals service integrity like 000 off nds which is in Porto zero.

Problem Number 13 Let's see, be the closed curve as the surfaces closes and f back to field to experience says that the line and the service entitle or equal. In other words, we can say that the lie integral off FB X equals the surface integral off the girl of the Vector Fee, Death and DS To evaluate the line integral. It is enough to evaluate the surface integral. It requires computing the curl of the vector field f which is equal to boy G m key partial derivative with respect to that partial derivative with respect to why and partial derivative with respect to X off X squared minus X squared. Why toe accident? This is equal to partial derivative. We respect. Why? Off to accident. Finest portion Derivative boy with respect. Oh, is it minus the partial derivative of negative off to accident with respect to X plus the partial derivative off X squared minus that squared with respect. Oh, said finally, the partial derivative of why with respect Toe X minus two partial derivative export whom I minus sweetie said forward to with respect to why the fourth this is equal to they go negative force that and zero. The output normal to the plane is in which is equal to negatives. Add X negative That boy and one which is equal toe 11 and one. The surface integral is zero negative for dessert and you will multiply one one on one. Yes, is equal to negative For that d s. The surface is the first obtained and hence zero X Why hands it are bigger than or equal to zero. Besides, by definitions that equals forward minus x minus y Before we can see that X why answered our necks than or equal fourth. We can now say that the surface integral off the curl of the vector field left and the s equals for multiply the double integral from zero till four and from the able till four minus y off four minus X minus y the x d y, which is equal It all four multiplies the integral from 0 to 4 off four x minus X squared over two minus y x substitute by our end conditions for minus y and zero de y, which is equal. Do wolf multiply the integral off from 0 to 4 off. Four would mind display minus four minus y squared for old squared over two minus short minus by why? Anyway, this is equal to with the integral from four multiplied internal from zero till food off. Why minus four squared over to do you work which is equal toe two months cry y miles fourth by 3/3 we substitute by our hand conditions for and zero together this integral is equal toe you.

Problem Number 13 Let's see, be the closed curve as the surfaces closes and f back to field to experience says that the line and the service entitle or equal. In other words, we can say that the lie integral off FB X equals the surface integral off the girl of the Vector Fee, Death and DS To evaluate the line integral. It is enough to evaluate the surface integral. It requires computing the curl of the vector field f which is equal to boy G m key partial derivative with respect to that partial derivative with respect to why and partial derivative with respect to X off X squared minus X squared. Why toe accident? This is equal to partial derivative. We respect. Why? Off to accident. Finest portion Derivative boy with respect. Oh, is it minus the partial derivative of negative off to accident with respect to X plus the partial derivative off X squared minus that squared with respect. Oh, said finally, the partial derivative of why with respect Toe X minus two partial derivative export whom I minus sweetie said forward to with respect to why the fourth this is equal to they go negative force that and zero. The output normal to the plane is in which is equal to negatives. Add X negative That boy and one which is equal toe 11 and one. The surface integral is zero negative for dessert and you will multiply one one on one. Yes, is equal to negative For that d s. The surface is the first obtained and hence zero X Why hands it are bigger than or equal to zero. Besides, by definitions that equals forward minus x minus y Before we can see that X why answered our necks than or equal fourth. We can now say that the surface integral off the curl of the vector field left and the s equals for multiply the double integral from zero till four and from the able till four minus y off four minus X minus y the x d y, which is equal It all four multiplies the integral from 0 to 4 off four x minus X squared over two minus y x substitute by our end conditions for minus y and zero de y, which is equal. Do wolf multiply the integral off from 0 to 4 off. Four would mind display minus four minus y squared for old squared over two minus short minus by why? Anyway, this is equal to with the integral from four multiplied internal from zero till food off. Why minus four squared over to do you work which is equal toe two months cry y miles fourth by 3/3 we substitute by our hand conditions for and zero together this integral is equal toe you.

Problem Number 15. The kernel of the factor field of F equals. I increase partial derivative with respect to that partial derivative with respect to why portion derivative with respect to X X squared minus y squared zed squared minus X squared and Y squared minus red square. This is equal to the partial derivative off. Why squared miners that squared respect? Why, minus the partial derivative was That's where minus X squared with respect was dead negative. Depart shin derivative Both. Why squared minus X squared or with respect to X minus Portion Derivative plus Tony. Partial derivative Off X squared minus y squared with us back to ex partial derivative off that squared minus X squared with respect to X minus the partial derivative X squared minus y squared with respect toe boy, this is equal to who I managed to zint zero and negative two X plus two. Why the normal vector to that equal? Zero is represented by n equals 00 on one. Using stoops low the surface integral or the curl of the vector field off F and G s equals. So why minus two Zint Jiro Negative two x plus two y want deploy 00 and one DS, which is ableto the double integral off from negative one Jews to sorry from negative one till one and from negative one till one off the negative two x plus tau Why the x d y, which is equal to the integration from negative one killed, one off the negative four x d x, which is able to we need to export to on dso secured by our end conditions to get final answer of this integral, which is equal toe is you.


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