Shape Operator of Sphere, Pseudosphere, and u^2-v^2 Saddle We require the use of the dot product, cross product and Eudlidean norm. Procedures are written for each of these.dp := proc(X,Y) X*Y+X*Y+X*Y; end: xp := proc(X,Y) local a,b,c; a := X*Y-X*Y; b := X*Y-X*Y; c := X*Y-X*Y; [a,b,c]; end: nrm := proc(X) sqrt(dp(X,X)); end: The following procedure gives the unit normal to an input surface X, where X is in paramaterized form and in square brackets.UN := proc(X) local Xu,Xv,Z,s; Xu := [diff(X,u),diff(X,u),diff(X,u)]; Xv := [diff(X,v),diff(X,v),diff(X,v)]; Z := xp(Xu,Xv); s := nrm(Z); simplify([Z/s,Z/s,Z/s],sqrt,symbolic); end: Surface Normal and the Shape Operator on the Sphere Given a parametrization expressed in terms of u and v, computes the tangent vectors, the surface normal and unit surface normal: X:=[r*cos(u)*cos(v),r*sin(u)*cos(v),r*sin(v)]; Xu := [diff(X, u), diff(X, u), diff(X, u)]; Xv := [diff(X, v), diff(X, v), diff(X, v)]; simplify(xp(Xu, Xv)); simplify(UN(X)); Computes negative the covariant derivative of the surface normal in the u direction and simplifies it: simplify(-diff(UN(X),u)); Computes negative the covariant derivative of the surface normal in the v direction and simplifies it: simplify(-diff(UN(X),v)); Surface Normal and the Shape Operator for the Pseudosphere Given a parametrization expressed in terms of u and v, computes the tangent vectors, the surface normal and unit surface normal: X:=[sech(u)*cos(v),sech(u)*sin(v),u-tanh(u)]; Xu := [diff(X, u), diff(X, u), diff(X, u)]; Xv := [diff(X, v), diff(X, v), diff(X, v)]; simplify(xp(Xu, Xv)); simplify(UN(X)); Computes negative the covariant derivative of the surface normal in the u direction and simplifies it: simplify(-diff(UN(X),u)); Xu := [diff(X, u), diff(X, u), diff(X, u)]; Xv := [diff(X, v), diff(X, v), diff(X, v)]; Computes negative the covariant derivative of the surface normal in the v direction and simplifies it: simplify(-diff(UN(X),v)); Plots and solves for the scalar function weights of xu and xv by using linear algebra reduction: plot3d(X, u=0..5, v=0..10,axes=box); with(LinearAlgebra):Mu:=Transpose(Matrix([Xu,Xv,simplify(-diff(UN(X),u))])); ReducedRowEchelonForm(Mu); Mv:=Transpose(Matrix([Xu,Xv,simplify(-diff(UN(X),v))])); ReducedRowEchelonForm(Mv); Surface Normal and the Shape Operator on the u^2 - v^2 Saddle Given a parametrization expressed in terms of u and v, computes the tangent vectors, the surface normal and unit surface normal: X:=[u,v,u^2-v^2]; Xu := [diff(X, u), diff(X, u), diff(X, u)]; Xv := [diff(X, v), diff(X, v), diff(X, v)]; simplify(xp(Xu, Xv)); simplify(UN(X)); Computes negative the covariant derivative of the surface normal in the u direction and simplifies it: simplify(-diff(UN(X),u)); Xu := [diff(X, u), diff(X, u), diff(X, u)]; Xv := [diff(X, v), diff(X, v), diff(X, v)]; Computes negative the covariant derivative of the surface normal in the v direction and simplifies it: simplify(-diff(UN(X),v)); Solves for the scalar function weights of xu and xv by using linear algebra reduction: with(LinearAlgebra):Mu:=Transpose(Matrix([Xu,Xv,simplify(-diff(UN(X),u))])); ReducedRowEchelonForm(Mu); Mv:=Transpose(Matrix([Xu,Xv,simplify(-diff(UN(X),v))])); ReducedRowEchelonForm(Mv); plot3d(X, u=-1/2..1/2, v=-1/2..1/2,axes=box); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUrZXhlY3V0YWJsZUdGNEYv