Paper 3, Section II, A

Vector Calculus | Part IA, 2014

The surface CC in R3\mathbb{R}^{3} is given by z2=x2+y2z^{2}=x^{2}+y^{2}.

(a) Show that the vector field

F(x)=(xyz)\mathbf{F}(\mathbf{x})=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)

is tangent to the surface CC everywhere.

(b) Show that the surface integral SFdS\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S} is a constant independent of SS for any surface SS which is a subset of CC, and determine this constant.

(c) The volume VV in R3\mathbb{R}^{3} is bounded by the surface CC and by the cylinder x2+y2=1x^{2}+y^{2}=1. Sketch VV and compute the volume integral

VFdV\int_{V} \nabla \cdot \mathbf{F} d V

directly by integrating over VV.

(d) Use the Divergence Theorem to verify the result you obtained in part (b) for the integral SFdS\int_{S} \mathbf{F} \cdot \mathbf{d} \mathbf{S}, where SS is the portion of CC lying in 1z1-1 \leqslant z \leqslant 1.

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