Website owner: James Miller
Point transformations. The system
1) u = u(x, y)
v = v(x, y)
represents a function that assigns to every number pair (x, y) another number pair (u, v). The number pairs (x, y) and (u, v) can be viewed as representing points in a plane and the system can be viewed as defining a point transformation that maps a point (x, y) in an xy-coordinate system into a point (u,v) in a uv-coordinate system. See Fig. 1. In the same way the system
u = u(x, y, z)
2) v = v(x, y, z)
w = w(x, y, z)
represents a function that assigns to every number triple (x, y, z) another number triple (u, v, w). The number triples (x, y, z) and (u, v, w) can be viewed as representing points in three-dimensional space and the system can be viewed as defining a point transformation that maps a point (x, y, z) in an xyz-coordinate system into a point (u, v, w) in a uvw-coordinate system. Generalizing on this idea the system of equations
u1 = u1(x1, x2, ... , xn)
3) u2 = u2(x1, x2, ... , xn)
um = um(x1, x2, ... , xn)
assigns a point (u1, u2, ... , um) in m-dimensional space to a point (x1, x2, ... , xn) in n-dimensional space. The domain is some specified point-set in n-dimensional space and the range is some point-set in m-dimensional space. It can be viewed as defining a point transformation from n-space into m-space.
Continuity and differentiability of point transformations. A point transformation is said to be continuous if the defining functions u1, u2, ... , un are continuous. It is said to be differentiable if these functions are differentiable. It is continuously differentiable if these functions have continuous partial derivatives.
One-to-one transformations. Let a point transformation map some region R into a region R'. If every point of R is mapped into a different point of R', no two points of R being mapped into the same point, the mapping is one-to-one. In a one-to-one mapping there is established a one-to-one correspondence between the points in R and R' with each point in region R being mapped into its correspondent in R'.
Inverse of a point transformation. A point transformation T can have an inverse transformation T-1 if and only if T maps in a one-to-one fashion. Let T be a one-to-one transformation mapping each point of a region R into some point in region R'. Then the inverse transformation T-1 maps each point of R' into that point in R that was imaged into it under transformation T. If a point transformation
u = u(x, y)
v = v(x, y)
is one-to-one then there exists an inverse transformation
x = x(u, v)
y = y(u, v)
that maps each point (u, v) back into its correspondent point (x, y).
Jacobian of a transformation. The Jacobian of the point transformation
u1 = f1(x1, x2, ... , xn)
u2 = f2(x1, x2, ... , xn)
un = fn(x1, x2, ... , xn)
Example. For the point transformation
u = f1(x, y)
v = f2(x, y)
the Jacobian is given by
Physical interpretation of the Jacobian. Consider the point transformation
u = u(x, y)
v = v(x, y)
Assume the transformation maps a small area ΔA into ΔA'. Then
where J is the Jacobian of the transformation.
Condition for a point transformation to be one-to-one. A point transformation is one-to-one at a point if the Jacobian does not vanish at the point. A point transformation is one-to-one in a specified region if the Jacobian of the transformation does not vanish within the region. If a transformation is one-to-one an inverse transformation exists. If an inverse exists, the Jacobian of the inverse transformation is the reciprocal of the Jacobian of the transformation.
Significance of the sign of the Jacobian. Consider a mapping at a point (x0, y0) where the Jacobian is not zero and the mapping is thus, at least locally, one-to-one. If C is a small closed curve encircling the point (x0, y0) in the xy plane the image of C will be a small closed curve C' encircling the point (u0, u0), the image of (x0, y0), in the uv plane. Let the point (x, y) goes around C in the counterclockwise direction. The image point (u, v) will then go around C'. If the Jacobian is greater than zero the image point (u, v) will go around C' in the same direction that point (x, y) goes around C. If the Jacobian is less than zero, it will go around in the opposite direction.