https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n− k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
{\displaystyle f(q_{1},q_{2},\ldots ,q_{n})=0,}
where the qi are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
{\displaystyle F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,}
where the πi are dimensionless parameters constructed from the qi by p = n − k dimensionless equations — the so-called Pi groups — of the form
{\displaystyle \pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},}
where the exponents ai are rational numbers (they can always be taken to be integers by redefining πi as being raised to a power that clears all denominators).
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n− k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
{\displaystyle f(q_{1},q_{2},\ldots ,q_{n})=0,}
where the qi are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
{\displaystyle F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,}
where the πi are dimensionless parameters constructed from the qi by p = n − k dimensionless equations — the so-called Pi groups — of the form
{\displaystyle \pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},}
where the exponents ai are rational numbers (they can always be taken to be integers by redefining πi as being raised to a power that clears all denominators).
Original post by golgiapparatus31
https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n− k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
{\displaystyle f(q_{1},q_{2},\ldots ,q_{n})=0,}
where the qi are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
{\displaystyle F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,}
where the πi are dimensionless parameters constructed from the qi by p = n − k dimensionless equations — the so-called Pi groups — of the form
{\displaystyle \pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},}
where the exponents ai are rational numbers (they can always be taken to be integers by redefining πi as being raised to a power that clears all denominators).
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n− k dimensionless parameters π1, π2, ..., πp constructed from the original variables. (Here k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.)
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
More formally, the number of dimensionless terms that can be formed, p, is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
{\displaystyle f(q_{1},q_{2},\ldots ,q_{n})=0,}
where the qi are the n physical variables, and they are expressed in terms of k independent physical units, then the above equation can be restated as
{\displaystyle F(\pi _{1},\pi _{2},\ldots ,\pi _{p})=0,}
where the πi are dimensionless parameters constructed from the qi by p = n − k dimensionless equations — the so-called Pi groups — of the form
{\displaystyle \pi _{i}=q_{1}^{a_{1}}\,q_{2}^{a_{2}}\cdots q_{n}^{a_{n}},}
where the exponents ai are rational numbers (they can always be taken to be integers by redefining πi as being raised to a power that clears all denominators).
Hi, please explain the concepts or stuff instead of copying stuff.
Original post by Eimmanuel
Hi, please explain the concepts or stuff instead of copying stuff.
Fully explained by the copypasta.
Have your own go at explaining concepts or stuff!
(edited 5 years ago)
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