A right circular cone has its apex at the origin and its lateral surface described

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Tensors and differential geometry

Invariance, Ch Differential forms and the exterior derivative

Problem

A right circular cone has its apex at the origin and its lateral surface described, for 0≤r≤R0\le r\le R and 0≤φ<2π0\le\varphi \lt 2\pi, by r(r,φ)=(rcos⁡φ, rsin⁡φ, hRr),\mathbf r(r,\varphi)=\left(r\cos\varphi,\ r\sin\varphi,\ \frac{h}{R}r\right), so that the base circle r=Rr=R sits at height z=hz=h. Take R=3 mR=3\,\mathrm m and h=4 mh=4\,\mathrm m. Consider the two-form ω=z dx∧dy\omega=z\,dx\wedge dy, the flux form of the field F=zz^\mathbf F=z\hat{\mathbf z}. ((a)) Using dx∧dy=r dr∧dφdx\wedge dy=r\,dr\wedge d\varphi (already established for plane polar coordinates), find the pullback r∗ω\mathbf r^*\omega to the (r,φ)(r,\varphi) parameter plane. ((b)) Integrate the pulled-back form over 0≤r≤R0\le r\le R, 0≤φ<2π0\le\varphi \lt 2\pi to obtain the flux of F\mathbf F through the lateral surface of the cone, oriented by the parameter order (r,φ)(r,\varphi). ((c)) Check the result against the surface-integral flux ∬F⋅(∂rr×∂φr) dr dφ\iint\mathbf F\cdot(\partial_r\mathbf r\times\partial_\varphi\mathbf r)\,dr\,d\varphi, held from the vector-calculus course.

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