Integration by Parts

By integrating both sides of this expression we obtain uv = jvdu + judv. 

Transpose the last term to the left-hand side  

The selection of the proper values of u and v is to be determined by trial. A little practice will enable one to make the right selection instinctively. The rule is that the integral Jv,du must be more easily integrated than the given expression. In dealing with Ex. (4), for instance, if we had taken u = dv = xdx, jv. du would have assumed the form x2e dxt which is a more complex integral than the one to be reduced. 

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Integrating by parts and leaving aside the non significarrt terms, the integral (3) can be handled to obtain... 

An integration by parts was used to deduce equation (A2.2.155) from equation (A2.2.154). Comparing the results for U and P, one finds that, just as for the classical gas, for ideal quanUim gases, also, the relation U = PFis satisfied. In the above results it was found that P = P(pp) and N)IV =n = n Cpp). In principle, one... 

If z = exp(pp) l, one can also consider the leading order quantum correction to the classical limit. For this consider tlie thennodynamic potential cOq given in equation (A2.2.144). Using equation (A2.2.149). one can convert the sum to an integral, integrate by parts the resulting integral and obtain the result ... 

Also, employ integration by parts to convert (plus. V= C ), yielding... 

Integration by parts (Green s theorem) of the second order term in Equation (2.90) gives the weak form of the problem as... 

In practice, in order to maintain the symmetry of elemental coefficient matrices, some of the first order derivatives in the discretized equations may also be integrated by parts. 

Here t = (—n2,ni) is the tangential unit vector at T. Integrating by parts, one can obtain the Green formula (Temam, 1983 Khludnev, Sokolowski, 1997)... 

Let us assume that kij = 0 on L. This enables us to integrate by parts in the second and the third terms of (2.231) and to obtain the inequality... 

The term corresponding to can be estimated easily since one can directly integrate by parts ... 

Integrating by parts on the left-hand side implies... 

In fact, we can integrate by parts on the right-hand side of (4.127), which gives... 

The formula (4.127) can be written in the form which does not contain the function 9. To show this we choose a ball B, (r) of radius r with the boundary T(r) such that 9 = 1 on Bx, r). In this case the integration by parts in (4.127) yields... 

We have obtained the Griffith formula (4.159). It is not difficult to show that the right-hand side of (4.159) does not depend on 9. To prove this, consider the difference between right-hand sides of (4.159) corresponding to any two functions 9i, 02- Let 9 = 9i — 92- We integrate by parts, which implies that the difference A between the right-hand sides of (4.159) evaluated for is equal to... 

We do not show the dependence of v,a,, f on t in (5.87). The integration by parts in the third term of the left-hand side of (5.87) can be done. Recall that satisfies the equation (5.68). As a result the following inequality is obtained ... 

Boundary conditions (5.81) can be taken into account here in order to integrate by parts in the left-hand side. Next we can integrate the inequality obtained in t from 0 to t. This implies... 

Integrate by parts in the fifth and sixth terms of the left-hand side of (5.152) taking into account the boundary conditions (5.149)-(5.151) and the Green formula like (5.138) for the domain fic- The penalty term is nonnegative and satisfy the equation (5.144). Hence the uniform in the s,5 estimate follows. 

The inclusion m G K can be proved by standard arguments. Note that the second boundary condition (5.142) and the conditions (5.143) are included in the identity (5.145). This means that it is possible to obtain these conditions by integrating by parts provided that the solution is sufficiently smooth. Actually, we can prove that the second condition (5.142) holds in the sense 77 / (F), but the arguments are omitted here. The theorem is proved. 

In so doing we have omitted the nonnegative term containing the penalty operator. Using the formula (5.181), the integration by parts can be done in the third and the fifth terms of the left-hand side of (5.189). Also, note that Mij satisfy equation (5.175). Integration of (5.189) in t from 0 to t results in the inequality... 

Taking into account the conditions (5.187), we first integrate by parts in the left-hand side of the inequality obtained and next we integrate in t. Simultaneously, the integration by parts in t is fulfilled. This gives the inequality... 

Notice that the right-hand side of (5.201) is the function bounded in (0, T) uniformly in 5. Let us integrate by parts in the left-hand side of (5.201) and use the fact that satisfy equation (5.175). Combining (5.201) with (5.200) we infer that... 

The second boundary condition (5.214) and the conditions (5.215) are involved in (5.218). This means that those conditions hold at any point of r, r, respectively, provided the solution v, rriij is smooth enough. The statement can be verified by integrating by parts. Theorem 5.7 is proved. 

This equation is integrated by parts to give the following equation... 

The Laplace transform of a derivative dy/dt is found by application of Equation (3-65) and integration by parts ... 

Using the kernel (250) in (243) with the initial condition (242) and integrating by parts, one obtains (PI)... 

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