Let C be the ellipse contained in the xy plane whose equation is 4x^2 + y^2 = 4, oriented clockwise. The force field F described by

shadsiei 2021-02-05 Answered

Let C be the ellipse contained in the xy plane whose equation is 4x2+y2=4, oriented clockwise. The force field F described by F(x,y,z)=x2i+2xj+z2k, moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: z=44x2y2,z0 (with ascending orientation, that is, the component in the z direction equal to 1) and assuming π=3.14, we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore W=0.
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that W=12.56.
c) We can apply Stokes' Theorem and conclude that W=6.28
d) We can apply Stokes' Theorem and conclude that W=12.56.

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Expert Answer

Obiajulu
Answered 2021-02-06 Author has 98 answers

Step 1
Given the force field,
F(x,y,z)=x2i+2xj+z2k
Also, C is the surface boundary of S:z=44x2y2,z0
Then, C is an ellipse
4x2+y2=4
Then, using the Stoke's Theorem, work done W is given by,
W=CFdr
=S(×F)dS...(1)
Here, ×F=(0,0,2)
Also, note that the orientation of the surface is clockwise,
Hence,
n=(0,0,1)
Step 2
Then, from equation(1),
W=S(0,0,2)(0,0,1)dS
=2SdS
=2×(π×1×2)
=12.566 units
Inverting the orientation of the surface S,
W=S(0,0,2)(0,0,1)dS
=2SdS
=2×(π×1×2)
=12.566 units
Thus, the correct option is,
b)Inverting the orientation of the surface S, we can apply Stoke's Theorem and conclude that W=12.56 units.

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