Functional Differential Geometry

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**a.** The rectangular coordinate equation for the Lemniscate of Bernoulli is

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Find the expression in polar coordinates.

**b.** Describe a helix space curve in both rectangular and cylindrical coordinates.

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(define-coordinates(up x y)R2-rect)(define-coordinates(up r theta)R2-polar);(defineR2-rect-chi(chart R2-rect)); R2-rect-chi ; generates the rectangle coordinates of a point.(defineR2-rect-chi-inverse(point R2-rect)); R2-rect-chi-inverse ; gets the abstract representation.(x(R2-rect-chi-inverse(up 'x0 'y0))); Function x ; gets the x coordinate ; of an (abstract-represented) point. ;(defineR2-polar-chi(chart R2-polar))(defineR2-polar-chi-inverse(point R2-polar))(x(R2-polar-chi-inverse(up 'r0 'theta0)))(r(R2-polar-chi-inverse(up 'r0 'theta0)))(r(R2-rect-chi-inverse(up 'x0 'y0)))(theta(R2-rect-chi-inverse(up 'x0 'y0)));(defineh(+(* x(square r))(cube y)))(defineR2-rect-point(R2-rect-chi-inverse(up 'x_0 'y_0)))(show-expression(h R2-rect-point))(show-expression(h(R2-polar-chi-inverse(up 'r_0 'theta_0))))

(show-expression((- r(* 2 'a(+ 1(costheta))))((point R2-rect)(up 'x 'y))))

; Ex 2.1 a(show-expression((-(square(+(square x)(square y)))(* 2(square 'a)(-(square x)(square y))))((point R2-rect)(up 'x 'y))))(show-expression((-(square(+(square x)(square y)))(* 2(square 'a)(-(square x)(square y))))((point R2-polar)(up 'r 'theta))))

; Ex 2.1 b(define-coordinates(up r theta z)R3-cyl)(define-coordinates(up x y z)R3-rect)(show-expression((-(up r z)(up 'R(* 'a theta)))((point R3-cyl)(up 'r 'theta 'z))))(show-expression((-(up r z)(up 'R(* 'a theta)))((point R3-rect)(up 'x 'y 'z))))

— Me@2022-12-10 10:29:59 AM

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2022.12.10 Saturday (c) All rights reserved by ACHK

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