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Jacobian is greater than zero the image point (u, v) will go around C' in the same direction that The image point (u, v) will then go around C'. Let the point (x, y) goes aroundĬ in the counterclockwise direction. If C is a small closedĬurve encircling the point (x 0, y 0) in the xy plane the image of C will be a small closed curve C'Įncircling the point (u 0, u 0), the image of (x 0, y 0), in the uv plane. The Jacobian is not zero and the mapping is thus, at least locally, one-to-one. Consider a mapping at a point (x 0, y 0) where Significance of the sign of the Jacobian. Inverse exists, the Jacobian of the inverse transformation is the reciprocal of the Jacobian of the If a transformation is one-to-one an inverse transformation exists. Transformation is one-to-one in a specified region if the Jacobian of the transformation does not Transformation is one-to-one at a point if the Jacobian does not vanish at the point.
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Where J is the Jacobian of the transformation.Ĭondition for a point transformation to be one-to-one. Consider the point transformationĪssume the transformation maps a small area ΔA into ΔA'. That maps each point (u, v) back into its correspondent point (x, y).
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Is one-to-one then there exists an inverse transformation Transformation T -1 maps each point of R' into that point in R that was imaged into it under
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Transformation mapping each point of a region R into some point in region R'. Transformation T -1 if and only if T maps in a one-to-one fashion. A point transformation T can have an inverse R being mapped into its correspondent in R'. In a one-to-one mapping there isĮstablished a one-to-one correspondence between the points in R and R' with each point in region Mapped into the same point, the mapping is one-to-one. If every point of R is mapped into a different point of R', no two points of R being Let a point transformation map some region R into a If these functions have continuous partial derivatives. Is said to be differentiable if these functions are differentiable. Transformation is said to be continuous if the defining functions u 1, u 2. mapping, point transformation, transformationĬontinuity and differentiability of point transformations. It can be viewed as defining a point transformation from n-space into m-space. The domain is some specified point-set in n-dimensional space and the range is some , u m) in m-dimensional space to a point (x 1, x 2. Generalizing on this idea the system of equationsĪssigns a point (u 1, u 2. Point (x, y, z) in an xyz-coordinate system into a point (u, v, w) in a uvw-coordinate system. 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 Represents a function that assigns to every number triple (x, y, z) another number triple (u, v, w). 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. Viewed as representing points in a plane and The number pairs (x, y) and (u, v) can be Number pair (x, y) another number pair (u, v). Represents a function that assigns to every