This short note gives the derivation of OLS estimators, proofs of linearity and unbiasedness as well the proof of minimum variances and also the standard error of sigma square from Gujarati.
Routine and advanced problems on systems of linear equations with an introduction to matrices.Descripción completa
Equations of mathematical physics
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ALternate derivation to Scherrer formula for X ray diffraction
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Derivation of the K epsilon Model in turbulence
Notes on the proper way to derive the equilibrium constant. Relevant for high school and university chemistry education.
Comment dimensionner un ouvrage de dérivation
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Derivation of Kirchoff’s Equations A. Diagram
U
O’
O
B. Kinetic Energy of the Liquid (See (Karamcheti, 1966) pg 302 for full derivation) The kinetic energy of the liquid is given by
∯ where is the velocity potential, is the gradient operator, and is the vector pointing normal to the surface of the 1
body. The integral is taken over just the surface of the body, and represents the kinetic energy of all fluid outside the body (the region outside the body extends to infinity).
C. Kinetic Energy of the Solid (See (Milne-Thomson, 1962) pg 523 for full derivation) The kinetic energy of the solid is given by
∰[ ]
2
where is the density of the solid, O is the origin of the inertially fixed reference frame, O’ is the origin of the body fixed frame, r is the vector from the origin O to O’, is the velocity of O’, V is the volume of the body, is the angular velocity of the body fixed frame.
D. Rate of Change of the Momentum The impulse (see (Karamcheti, 1966) pg 247 for full definition) is an infinitesimally small change in linear and angular momentum. Assume the total kinetic energy of the system given by
1 Ball
3
is initially zero, and that the motion of the fluid is due completely to the motion of the body. Also assume that the body is brought from rest into motion by an impulse wrench (an impulse wrench is a force moment couple (F,L) applied for an infinitesimal amount of time applied to the body). The rate of change of the momentum assuming all vectors are given in the inertial reference frame (denoted by subscript i) is given by
4 5
are identically Eulers equations of motion where is the linear momentum and is the angular momentum taken about O (note: derivatives are taken with respect to the inertial frame). Assuming the origin O and O’ are initially coincident and that the vectors and are given in the body frame (denoted by the subscript b), the rate of change of the unit vectors in the body frame ( ) needs to be taken into account via the transport theorem given by
̂ ̂ ̂
6
This leads to the equations of motion of the following form
where the term comes from the fact that the angular momentum is taken about the point O. If it had been taken about O’ this term would equal zero (note: some may refer to angular momentum defined in this way as “moment of momentum”, and “angular momentum” when the momentum is measured about O’).
E. Kirchhoff’s Equations The work done by a force-wrench over an infinitesimal time (the impulse) equals the change in kinetic energy of 1 the system . Using Eulers Theorem of Homogeneious Functions of degree 2 shown below
where T is an arbitrary scalar function of and , and and are two independent vectors, one can show that 9
10
Substituting the relations given by 10 into equations 7 and 8, one arrives at Kirchhoff’s Equatios of Motion for a rigid body in an infinite fluid shown below
where W is the work done one the system and is
a homogeneous linear function.
2 Ball
where T is the total kinetic energy. F. Referances Karamcheti, K. (1966). Principles of Ideal-Fluid Aerodynamics. Malabar: Krieger Publishing Company. Milne-Thomson, L. M. (1962). Theoretical Hydrodynamics (Fourth Edition). London: Macmillan & Co LTD.
