Relative scalar: Difference between revisions

Revision as of 22:23, 19 April 2011

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===Elementary example===

Suppose the temperature in a room is given in terms of the function <math>f(x,y,z)= 2 x + y + 5</math> in Cartesian coordinates (x,y,z) and we desire the function in cylindrical coordinates (r,t,z'h) . The two coordinate systems are related by the following sets of equations:

The two coordinate systems are related by the following sets of equations:

:<math> r = \sqrt{x^2 + y^2}</math>

:<math> t = \arctan{(y/x)}</math>

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Suppose that one wishes to ''integrate'' these functions over "the room", which we will denote by <math>D</math>. (Yes, integrating temperature is strange but that's partly what's to be shown.) Suppose the region <math>D</math> is given

in cylindrical coordinates as r from [0,2], t from [0,\pi/2] and h from [0,2] (that is, our room is a quarter slice of a cylinder of radius and height 2). The value of the integral of <math>\bar{f}</math> over <math>D</math> is <math>2 (6 + 5 \pi)</math>. If the integral of <math>f</math> is made over the same region, the result is <math>64/3 + 20 \pi</math>. They are not equal! The integral of a ordinary scalar depends on the coordinate system used.

==Scalar density==