Curvilinear coordinates are a coordinate system for the Euclidean space based on some transformation that converts the standard Cartesian coordinate system to a coordinate system with the same number of coordinates in which the coordinate lines are curved. In the two-dimensional case, instead of Cartesian coordinates x and y, e.g., p and q are used: the level curves of p and q in the xy-plane. Required is that the transformation is locally invertible (a one-to-one map) at each point. This means that one can convert a point given in one coordinate system to its curvilinear coordinates and back.
Depending on the application, a curvilinear coordinate system may be simpler to use than the Cartesian coordinate system. For instance, a physical problem with spherical symmetry defined in R3 (e.g., motion in the field of a point mass/charge), is usually easier to solve in spherical polar coordinates than in Cartesian coordinates. Also boundary conditions may enforce symmetry. One would describe the motion of a particle in a rectangular box in Cartesian coordinates, whereas one would prefer spherical coordinates for a particle in a sphere.
Many of the concepts in vector calculus, which are given in Cartesian or spherical polar coordinates, can be formulated in arbitrary curvilinear coordinates. This gives a certain economy of thought, as it is possible to derive general expressions—valid for any curvilinear coordinate system—for concepts as gradient, divergence, curl, and the Laplacian. Well-known examples of curvilinear systems are polar coordinates for R2, and cylinder and spherical polar coordinates for R3.
The name curvilinear coordinates, coined by the French mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. While a Cartesian coordinate surface is a plane, e.g., z = 0 defines the x-y plane, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere in R3—which obviously is curved.
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Q. Using the steps above, construct scatterplots and identify the relationship (positive linear, negative linear, curvilinear, or no relationship) for the following sets of data. Set #1 Set #2 Set #3 Set #4 ( 3, 50) ( 9, 10) (10, 20) (56, 54) ( 6, 40) ( 7, 2) ( 3, 3) (77, 81) (12, 28) ( 1, 5) (18, 2) (67, 11) ( 5, 38) (12, 14) (11, 19) (27, 56) ( 7, 20) ( 3, 0) (19, 1) (76, 67) ( 6, 35) ( 6, 8) ( 9, 19) (16, 71) ( 4, 45) ( 9, 7) ( 5, 12) (93, 68) ( 9, 25) (10, 11) ( 15, 11) (33, 31) ( 8, 30) (10, 5) ( 1, 2) (90, 67) ( 7, 26) ( 2, 1) (20, 1) (52, 32)
Asked by lastgo58 - Sun Jul 9 16:12:20 2006 - - 2 Answers - 0 Comments
A. have you tried to use excel to plot the 4 sets of data (4 lines)? by the way, what does the instruction say? what steps are you talking about???
Answered by Alice Lou - Sun Jul 9 21:36:17 2006
