How do you find a parametric representation?

jo

Consider the curve C given by the intersection of the surfaces y = x^2 - z^2 and x^2 + z^2 = 1

a) Find parametric representations of the curve C

b) If F=<zx^2,x^3,xz> find the integral of F.dr

c) Find the curl of this vector field. Integrate curlF.dS where S is the surface y=x^2-z^2 bounded by C

leo-allen_60bb864052dbf

x^2+z^2=1 and y=y is a cylinder , so the curve is

x^2+z^2 =1 y= x^2 -z^2

At plane XZ ( Region) take cylindrical coordinates

x= RcosT

z=RsinT R=1 ( Radius of the circle at XZ plane)

y= R^2(cos^2 T - sin^2T) , ie , y= R^2 cos 2T

then

x =cosT

z=sinT

y= cos2T ( Parametric)

r= cos T i+ sinTj+ cos2T k ( Position )

dr = dr/dT dT = -sinT, cosT, -2sin2T ) dT

F= (sinTcos^2T, cos^3T , cosTsinT) ( cosTsinT= sin2T/2)

INT Fdot dr= INT ( -sin^2Tcos^2T+ cos^3TcosT - sin^2 2T) dT 0<T<2pi

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