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Unformatted text preview: MATH 230 VECTOR CALCULUS AND ANALYSIS SECTION 3 5 36. (3 points) Find a formula for S F d S when S is given by x = k ( y,z ) and n is the unit normal that points forward (that is, toward the viewer when the axes are drawn in the usual way). Solution: S has the following parametrisation in terms of y and z , and associated tangent and normal vectors: r ( x,y,z ) = k ( y,z ) ,y,z , r y = k y , 1 , , r z = k z , , 1 , r y r z = 1 ,- k y ,- k z . Note that this is the correct orientation since the normal vector has positive x-component. Thus if the parameter domain is D , we have S F d S = D F ( r ( y,z )) ( r y r z ) dA = D P- Q k y- R k z dA. Section 17.8 4. (3 points) Use Stokes Theorem to evaluate S curl F d S , where F ( x,y,z ) = x 2 y 3 z i + sin( xyz ) j + xyz k , and S is the part of the cone y 2 = x 2 + z 2 that lies between the planes y = 0 and y = 3, oriented in the direction of the positive y-axis....
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