Paper 3 , Section I, A

Vector Calculus | Part IA, 2007

(i) Give definitions for the unit tangent vector T^\hat{\mathbf{T}} and the curvature κ\kappa of a parametrised curve x(t)\mathbf{x}(t) in R3\mathbb{R}^{3}. Calculate T^\hat{\mathbf{T}} and κ\kappa for the circular helix

x(t)=(acost,asint,bt),\mathbf{x}(t)=(a \cos t, a \sin t, b t),

where aa and bb are constants.

(ii) Find the normal vector and the equation of the tangent plane to the surface SS in R3\mathbb{R}^{3} given by

z=x2y3y+1z=x^{2} y^{3}-y+1

at the point x=1,y=1,z=1x=1, y=1, z=1.

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