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David W. Sutherland _________________________________________________________ dws3963

Week 4____________________________________________________________________3.4.3

http://hotard108.tripod.com/311hwk4a.html

Consider the function defined by:

> with(linalg):x=cosh(u)*cos(v);y=sinh(u)*sin(v);

Warning, new definition for norm

Warning, new definition for trace

Put this in vector form:

> A:=vector([cosh(u)*cos(v),sinh(u)*sin(v)]);

Note that the Jabcobian is given by

> a:=matrix(2,2,[dx/du,dx/dv,dy/du,dy/dv]);

This can be calculated by hand, but in this case it is easier to use Maple.

> jac:=jacobian(A, [u,v]);

Now substitute u=1 and v=0 into the jacobian to find the differential matrix. This gives the differential:

> differential:=matrix(2,2,[sinh(1),0,0,sinh(1)]);

> evalf(%);

b. Find the affine approximation to F at Uo

This approximation is given by F(u)=f(Uo)+f '(Uo)(U-Uo). Note that the f '(Uo) term is the differential calculated in a) and f(Uo) is given by u=0 and v=1 subbed into the original function.

> F(Uo):=matrix(2,1,[cosh(1)*cos(0),sinh(1)*sin(0)]);

> Fprime(Uo):=matrix(2,2,[sinh(1),0,0,sinh(1)]);

So, the approximation is:

> F(u):=F(Uo)+Fprime(Uo)*(u-matrix(2,1,[1,0]));

where "u" is a 2 X 1 matrix close to [1 0].