M E 422 Lecture 7: L7a-Pi theorem
14 Sep 2016
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I n addition, if dimensionless groups comparing two physical effects were considered, then f may acquire a simpler form. For instance, one could obtain via suitable groups. , x y)) where the dependance on v and w has evaporated when defining the groups. To answer this crucial question, we consider homogeneous functions in what follows: homogeneous functions: a function f(x,y) is homogeneous of order n if f (( ,x y)) >>> f (( Y)) n f (( ,x y)) (1) Consider the equation of a circle in cartesian coordinates. 2 y and the function f(x,y)=x2 +y2 is homogenous of order 2: differential of homogeneous functions: this is the important property towards understanding the independence of dimensionless parameters. Consider eqs 1 and 2 for a given value of : N f (( ,x y)) (3: physical laws are homogeneous with regard to units: Consider a physical law relating z,x and y:
