The Jacobian

The Jacobian of a transformation maps tangent vectors in the uv -plane to tangent vectors in the xy -plane. Indeed, the Jacobian can be used to approximate a coordinate transformation locally, which in turn implies that Jacobians are important in a wide variety of applications. In this worksheet, we explore the Jacobian and some of its applications.

> restart:with(linalg):with(plots):with(plottools):with(student):

# matsubs(eqs,matexpr) uses the list of equations eqs to
# substitute into the coefficients of the matrix matexpr

matsubs:=proc(eqs::list,matexpr::matrix)
local i,j,ii,jj,matexpn;
ii:=rowdim(matexpr);
jj:=coldim(matexpr);
matexpn := matrix(ii,jj);
for i from 1 to rowdim(matexpr) do
for j from 1 to coldim(matexpr) do
matexpn[i,j] := subs(eqs,matexpr[i,j]);
end do
end do;
evalm(matexpn)
end proc:

>

The Jacobian Matrix

The Jacobian Determinant

The Newton-Rhaphson Method

Exercises