Change of variable

X is an n-vector of n independent normal standardized variables, i.e., with zero mean, unit variances and null covariances. Find (i) the density function and (ii) the covariance matrix of the vector Y given by 

Using equation (4.2.29) that gives the density function of a dependent variable, we obtain 

Application of basic rules of matrix manipulation gives 

Applying the linearity property of the expectation to the change of variable given by equation (4.2.29), we get 

Multiplying the last two equations and using the definition of the covariance matrix, we obtain 

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We will assume in this article that the system is time-reversible, so T(p) = T —p). Dichotomic Hamiltonians arise from elementary particle models, the simplest nontrivial class of conservative systems. Moreover, even seemingly more complex systems can usually be written in the dichotomic form through change of variables or introduction of additional degrees of freedom. 

The same change of variables can also be made in equations (11.18), after which eauation (11.20) can be used to eliminate dp/dX, with the result... 

This unattractive integral is readily solved by introducing a change of variable... 

To justify the passage to the limit in the relations obtained from (2.127) by a change of variables, we use the convergence (2.128) and the statement analogous to Lemma 2.1. The limiting function % = U,u) is a solution of the variational inequality (2.100) with the function from (2.126), that is X = Xtl>- Finally, it is easy to verify that... 

Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. 

Change of Variable This substitution is basically the same as previously indicated for indefinite integrals. However, for definite integrals, the limits of integration must also be changed i.e., forx = ( )(t). 

The analytical methods for solving the Fourier equation, in which Q and T are functions of the spatial co-ordinate and time, include a change of variables by combination, and in the more general case the use of Laplace U ansforms. 

To find the temperature distribution, the first procedure requires a change of variables to the variable, z, defined by the equation... 

Since/is a tangential force, only the vector projection of the area is effective. Integrating these forces from 0 to Ti/2,f s given by equation 9.1-1 which is more easily done by a change of variables giving/= p r, and the stress is j Equation 9.1-2 gives... 

Now when we substitute U = f iv) into our equation, and introduce a change of variable to reflect the need to capture the very rapid changes of... 

A change of variables, from qi, pi ) to ( Qi, Pi ), is called a canonical transformation, if the transformed equations of motion have the same form as that given in equation 4,36, but with H = H( Qi, Pi )- In particular, if there exists a canonical transformation such that % = H Pi,P2,.., F ) only, then the equations... 

It should now be clear that the problem of calculating pY in terms oipx is the same as that of making the change of variable 17 = ( ) in the right-hand integral appearing in (3-42). This relationship can be written down explicitly if certain assumptions are made about the function . For example, if is a monotone, differentiable function (which implies that the inverse function < 1 defined by -1[ (f)] = exists) and if ( — co) = — co and (oo) = co, then... 

An example of this change-of-variable technique where is not monotonic arises when we choose a quadratic device, (x) = x2. Equation (3-42) is valid for any , so we can write... 

The next example will illustrate a different technique for solving change of variable problems. The background of the problem is the... 

Equation (3-104) (sometimes called the stationarity property of a probability density function) follows from the definition of the joint distribution function upon making the change of variable t = t + r... 

We now make the change of variable = + ij in the last integral to obtain the result... 

The changeover to thermodynamic activities is equivalent to a change of variables in mathematical equations. The relation between parameters and a. is unambiguous only when a definite value has been selected for the constant p. For solutes this constant is selected so that in highly dilute solutions where the system p approaches an ideal state, the activity will coincide with the concenttation (lim... 

We next make the following conventional change of variables... 

A particularly simple case is the study of a fiber in the slit-focus camera, if the fiber is stretched out along the slit direction [31,62,63]. In this case the transversal structure according to Bonart [16] (cf. Sect. 8.4.3) is directly measured, as is established by change of variables S —> s, (fiber parallel to the slit) s2,s —> S 2 (fiber symmetry assumed)... 

Notice the change of variables Guinier s law is applied to the Fourier transform of the peak as a function of the order (h) of the peak at fixed positions r. 

When both IxQ and x I are large a different approach to eq. (25) is needed. uThe change of variables t = (l-s)/u gives... 

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