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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
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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
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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