## File: dimen.dem

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maxima 5.6-17
 `1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465666768` ``````/* First load the necessary file: */ LOAD("dimen.mc")\$ /* It is conjectured that for thermistors there is a physical relationship between the voltage drop, current, ambient temperature, room-temperature resistance, convective heat transfer coefficient, and a constant, BETA, having the dimension of temperature. First, to see if the dimension of BETA is already known: */ GET(BETA, 'DIMENSION); /* It is not. To establish it: */ DIMENSION(BETA=TEMPERATURE); /* To automatically determine a set of dimensionless variables sufficient to characterize the physical relation: */ NONDIMENSIONALIZE([VOLTAGE, CURRENT, TEMPERATURE, RESISTANCE, HEATTRANSFERCOEFFICIENT, BETA]); /* We learn that the relation may be expressed as a function of only the above 3 variables rather than a function of the six physical quantities. Evidently dimensions were preestablished for all but the last of these particular input quantities, but an appropriate error message would have informed us if this were not so. An extensive set of dimensions have been prestablished, as may be seen from the listing of DIMEN >. As another example, there is thought to be a relation between the viscosity, average velocity, molecular mass, and repulsion coefficient of a gas. The repulsive force between two molecules is believed to be of the form K/DISTANCE^N, with unknown N, so K must have the following dimensions: */ DIMENSION(K=MASS*LENGTH^(N+1)/TIME^2) \$ /* To get the computation time in milliseconds to be printed automatically: */ CPUTIME: TRUE \$ /* To do a dimensional analysis of the gas viscosity problem: */ NONDIMENSIONALIZE([VISCOSITY, K, MASS, VELOCITY]); /* The physical relation must be expressible as a function of this one dimensionless variable, or equivalently, this variable must equal a constant. Consequently, physical measurements may be used to determine N. It turns out to be in the range 7 to 12 for common gases. As a final example, suppose that we conjecture a relation between the deflection angle of a light ray, the mass of a point mass, the speed of light, and the distance from the mass to the point of closest approach: */ NONDIMENSIONALIZE([ANGLE, MASS, LENGTH, SPEEDOFLIGHT]); /* We learn that there cannot be a dimensionless relation connecting all of these quantities and no others. Let us also try including the constant that enters the inverse-square law of gravitation: */ NONDIMENSIONALIZE([ANGLE, MASS, LENGTH, SPEEDOFLIGHT, GRAVITYCONSTANT]); /* Altermatively, for astrophysics problems such as this,we may prefer to use a dimensional basis in which the gravity constant is taken as a pure number, eliminating one member from our dimensional basis: */ %PURE: CONS(GRAVITYCONSTANT, %PURE); /* Note that the latter two of the above constants are pure numbers by default, respectively eliminating TEMPERATURE and CHARGE from the basis, but the user may include all five of TEMPERATURE, CHARGE, MASS, LENGTH, and TIME in the basis by resetting %PURE to []. Altermatively, the user may wish to include SPEEDOFLIGHT in %PURE for relativistic problems or PLANCKSCONSTANT for quantum problems. For dimensional analysis it doesn't really matter which basis member is eliminated by each pure constant, but in fact the latter two respectively eliminate LENGTH and TIME, whereas GRAVITYCONSTANT eliminates MASS. To proceed with our analysis: */ NONDIMENSIONALIZE([ANGLE, MASS, LENGTH, SPEEDOFLIGHT]); ``````