UVxvL0xpYnJhcnkvRnJhbWV3b3Jrcy9NYXBsZS5mcmFtZXdvcmsvVmVyc2lvbnMvMTEvYmluLkFQUExFX1BQQ19PU1g2Ig==

Common Code EDotProd:=(x,y)->LinearAlgebra[DotProduct](x,y,conjugate=false): ENorm:=x->LinearAlgebra[Norm](x,2,conjugate=false): FirstFF:=proc(x,u,v) local xu, xv, E, F, G, g; xu:= map(diff,x,u); xv:=map(diff,x,v); E:=EDotProd(xu,xu); F:=EDotProd(xu,xv); G:=EDotProd(xv,xv); g:=Matrix( [ E, F, F, G] ,2 ,2 ); return simplify(g) end: UnitNormal:=proc(x,u,v) local xu, xv, N, lenN ; xu:= map(diff,x,u); xv:=map(diff,x,v); N := simplify(LinearAlgebra[CrossProduct](xu,xv)); lenN := simplify( sqrt(EDotProd(N,N))); return simplify( N/lenN ); end: SecondFF:=proc(x,u,v) local U, xu, xv, xuu, xuv, xvv, l, m, n, A; xu:= map(diff,x,u); xv:=map(diff,x,v); xuu:=map(diff,xu,u); xuv:=map(diff,xu,v); xvv:=map(diff,xv,v); U:=UnitNormal(x,u,v); l:=EDotProd(xuu,U); m:=EDotProd(xuv,U); n:=EDotProd(xvv,U); A:=Matrix( [ l, m, m, n] ,2 ,2 ); return simplify(A) end: Shape:=proc(x,u,v) local g, A; g:=FirstFF(x,u,v); A:=SecondFF(x,u,v); return simplify(LinearAlgebra[MatrixInverse](g).A); end: Problem 4. GaussCurvature:=proc(x,u,v) local g, A; g:=FirstFF(x,u,v); A:=SecondFF(x,u,v); return simplify(LinearAlgebra[Determinant](A)/LinearAlgebra[Determinant](g)); end: MeanCurvature:=proc(x,u,v) local g, A; g:=FirstFF(x,u,v); A:=SecondFF(x,u,v); return simplify(1/2*LinearAlgebra[Trace](LinearAlgebra[MatrixInverse](g).A)); end: x:=<u,v,sqrt(R^2-u^2-v^2)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpJ1IkXDY=

GaussCurvature(x,u,v);

KiQpSSJSRzYiIiIjISIi

MeanCurvature(x,u,v) assuming u^2+v^2<R^2, R>0;

LCQqJEkiUkc2IiEiIkYm

Problem 5 Oprea 3.2.13 x:=<u,v,u^2/a^2+v^2/b^2>;

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

GaussCurvature(x,u,v);

LCQqKiIiJSIiIilJImFHNiIiIidGJSksKComKUYnRiRGJSlJImJHRihGJEYlRiUqKEYkRiUpSSJ1R0YoIiIjRiVGLkYlRiUqKEYkRiUpSSJ2R0YoRjNGJUYtRiVGJUYzISIiKUYvRilGJUYl

testeq(%,1/(4*a^2*b^2*(u^2/a^4+v^2/b^4+1/4)^2));

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Oprea 3.2.14 x:=<u,v,u^2/a^2-v^2/b^2>;

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

GaussCurvature(x,u,v);

LCQqKiIiJSIiIilJImFHNiIiIidGJSksKComKUYnRiRGJSlJImJHRihGJEYlRiUqKEYkRiUpSSJ1R0YoIiIjRiVGLkYlRiUqKEYkRiUpSSJ2R0YoRjNGJUYtRiVGJUYzISIiKUYvRilGJUY3

testeq(%,-1/(4*a^2*b^2*(u^2/a^4+v^2/b^4+1/4)^2));

SSV0cnVlRyUqcHJvdGVjdGVkRw==

Oprea 3.2.15 x:=<u*cos(v),u*sin(v),b*v>;

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

GaussCurvature(x,u,v);

LCQqJiksJiokKUkiYkc2IiIiIyIiIkYrKiQpSSJ1R0YpRipGK0YrRiohIiJGJ0YrRi8=

MeanCurvature(x,u,v);

IiIh

Oprea 3.2.16 x:=<u,v,u*v>;

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

GaussCurvature(x,u,v);

LCQqJCksKCIiIkYmKiQpSSJ2RzYiIiIjRiZGJiokKUkidUdGKkYrRiZGJkYrISIiRi8=

MeanCurvature(x,u,v);

LCQqKEkidUc2IiIiIkkidkdGJUYmKSwoRiZGJiokKUYnIiIjRiZGJiokKUYkRixGJkYmIyIiJEYsISIiRjE=

Problem 6 RuledGauss:=proc(beta,delta,u,v) betap:=map(diff,beta,u); deltap:=map(diff,delta,u); bpxd:=LinearAlgebra[CrossProduct](betap,delta); dpxd:=LinearAlgebra[CrossProduct](deltap,delta); W:=ENorm(bpxd+v*dpxd); return -simplify(EDotProd(betap,dpxd)^2/W^4); end;

Warning, `betap` is implicitly declared local to procedure `RuledGauss`

Warning, `deltap` is implicitly declared local to procedure `RuledGauss`

Warning, `bpxd` is implicitly declared local to procedure `RuledGauss`

Warning, `dpxd` is implicitly declared local to procedure `RuledGauss`

Warning, `W` is implicitly declared local to procedure `RuledGauss`

Zio2JkklYmV0YUc2IkkmZGVsdGFHRiVJInVHRiVJInZHRiU2J0kmYmV0YXBHRiVJJ2RlbHRhcEdGJUklYnB4ZEdGJUklZHB4ZEdGJUkiV0dGJUYlRiVDKD44JC1JJG1hcEclKnByb3RlY3RlZEc2JUklZGlmZkdGNDkkOSY+OCUtRjM2JUY2OSVGOD44Ji0mSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YlRjQ2I0ktQ3Jvc3NQcm9kdWN0R0YlNiRGMUY9PjgnLUZBNiRGOkY9PjgoLUkmRU5vcm1HRiU2IywmRj8iIiIqJjknRlJGSUZSRlJPLCQtSSlzaW1wbGlmeUc2JEY0RkQ2IyomLUkpRURvdFByb2RHRiU2JEYxRkkiIiNGTSEiJSEiIkYlRiVGJQ==

Part b. beta:=<u,0,0>; delta:=<0,1,u>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIoT1hNKg==

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpM2t1OA==

RuledGauss(beta,delta,u,v);

LCQqJCwoIiIiRiUqJEkidkc2IiIiI0YlKiRJInVHRihGKUYlISIjISIi

GaussCurvature(beta+v*delta,u,v);

LCQqJCwoIiIiRiUqJEkidkc2IiIiI0YlKiRJInVHRihGKUYlISIjISIi

Part c. beta:=<p1,p2,p3>;delta:=<delta1(u),delta2(u),delta3(u)>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpPz5HNw==

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpI2YqZTU=

RuledGauss(beta,delta,u,v);

IiIh

GaussCurvature(beta+v*delta,u,v);

IiIh

Part d. alpha:=<beta1(u),beta2(u),beta3(u)>;delta:=<d1,d2,d3>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpSyJwPyI=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpUydvNCI=

RuledGauss(beta,delta,u,v);

IiIh

GaussCurvature(beta+v*delta,u,v);

IiIh

Part e. beta:=<0,0,b*u>;delta:=<a*cos(u),a*sin(u),0>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIoIT14KCo=

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIob3I6Kg==

ruledGauss(beta,delta,u,v);

LCQqJkkiYkc2IiIiIywoKiRGJCIiJSIiIiooSSJhR0YlRiZJInZHRiVGJkYkRiZGJiomRixGKUYtRilGKiEiIkYv

GaussCurvature(beta+v*delta,u,v);

LCQqJkkiYkc2IiIiIywmKiRGJEYmIiIiKiZJInZHRiVGJkkiYUdGJUYmRikhIiMhIiI=

Problem 7 We just show here that F=0 and m=0 for a surface of revolution. See solutions for the rest of the problem. x:=<f(u)*cos(v),g(u),f(u)*sin(v)>;

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

FirstFF(x,u,v);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKCFHWCYp

SecondFF(x,u,v);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKFMqcCYp

Problem 8 x:=<u,c*cosh(u/c)*cos(v),c*cosh(u/c)*sin(v)>;

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g:=FirstFF(x,u,v);

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A:=SecondFF(x,u,v);

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k_mu=simplify(A[1,1]/g[1,1]) assuming c>0, u::real;

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k_pi:=simplify(A[2,2]/g[2,2]) assuming c>0, u::real;

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GaussCurvature(x,u,v);

LCQqJilJImNHNiIiIiMhIiIpLUklY29zaEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjKiZJInVHRiYiIiJGJUYoIiIlRihGKA==

MeanCurvature(x,u,v);

IiIh

Problem 9. g:='g';x:=<u*cos(v),g(u),u*sin(v)>;

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MeanCurvature(x,u,v);

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num:=op(2,%);

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dsolve({num=0,g(0)=1,D(g)(0)=0},g(u));

Ly1JImdHNiI2I0kidUdGJSIiIg==

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