Problem 4 UnitNormal:=proc(x,u,v) local x_u, x_v, xu_cross_xv, Nsize,U; x_u:=map(diff,x,u); x_v:=map(diff,x,v); xu_cross_xv:=simplify(LinearAlgebra:-CrossProduct(x_u,x_v)) assuming real; Nsize := simplify(sqrt(LinearAlgebra:-DotProduct(xu_cross_xv,xu_cross_xv))) assuming real; U:= simplify(xu_cross_xv/Nsize) assuming real; return U; end; x:=<v*cos(u),v*sin(u),b*u>;

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UnitNormal(x,u,v);

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Problem 5 FundamentalForms:=proc(x,u,v) local x_u, x_v, xu_cross_xv, Nsize,U, x_uu, x_uv, x_vv, E, F, G, l, m, n; x_u:=simplify(map(diff,x,u)) assuming real; x_v:=simplify(map(diff,x,v)) assuming real; x_uu:=simplify(map(diff,x_u,u)) assuming real; x_uv:=simplify(map(diff,x_u,v)) assuming real; x_vv:=simplify(map(diff,x_v,v)) assuming real; xu_cross_xv:=simplify(LinearAlgebra:-CrossProduct(x_u,x_v)) assuming real; Nsize := simplify(sqrt(LinearAlgebra:-DotProduct(xu_cross_xv,xu_cross_xv))) assuming real; U:= simplify(xu_cross_xv/Nsize) assuming real; E:=simplify(LinearAlgebra:-DotProduct(x_u,x_u)) assuming real; F:=simplify(LinearAlgebra:-DotProduct(x_u,x_v)) assuming real; G:=simplify(LinearAlgebra:-DotProduct(x_v,x_v)) assuming real; l:=simplify(LinearAlgebra:-DotProduct(x_uu,U)) assuming real; m:=simplify(LinearAlgebra:-DotProduct(x_uv,U)) assuming real; n:=simplify(LinearAlgebra:-DotProduct(x_vv,U)) assuming real; return < <E| F>, <F | G>>, < <l|m>,<m|n>>; end: Problem 6 x:=<cos(u)*sin(v),sin(u)*sin(v),cos(v)>;

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FundamentalForms(x,u,v) assuming 0<v, v<Pi;

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a:=%[1]

Warning, inserted missing semicolon at end of statement

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a[1][1]

Warning, inserted missing semicolon at end of statement

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Problem 8 Curvatures:=proc(x,u,v) local FI, FII, detI, K, H; (FI,FII):=FundamentalForms(x,u,v); detI:=simplify(LinearAlgebra:-Determinant(FI)) assuming real; K:=simplify(LinearAlgebra:-Determinant(FII)/detI ) assuming real; H:=simplify(1/2*(FI[1][1]*FII[2][2]+FI[1][1]*FII[2][2]-2*FI[1][2]*FII[1][2])/detI) assuming real; return K, H; end: x:=<cos(u)*v,sin(u)*v,b*u>;

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(K,H):=Curvatures(x,u,v);

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K;

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H;

IiIh

Problem 9 Shape:=proc(x,u,v) local forms, FI, FII; forms:=FundamentalForms(x,u,v); FI:=forms[1]; FII:=forms[2]; return simplify(LinearAlgebra:-MatrixInverse(FI).FII) assuming real; end: Problem 10 x:= <u,v,u*v>;

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Shape(x,u,v);

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