Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.9 STATIONARY VALUES UNDER CONSTRAINTS we findz=− 2 x. Substituting these values into the first constraint,x^2 +y^2 +z^2 =1,we ...

PARTIAL DIFFERENTIATION
A system contains a very large numberNof particles, each of which can be in any ofR energy levels with ...

5.10 ENVELOPES We now have the general form for the distribution of particles amongst energy levels, but in order to determine t ...

PARTIAL DIFFERENTIATION P P 1 P 2 x y f(x, y, α 1 )=0 f(x, y, α 1 +h)=0 Figure 5.4 Two neighbouring curves in thexy-plane of the ...

5.10 ENVELOPES his made arbitrarily small, so thatP→P 1 , the three defining equations reduce to two, which define the envelope ...

PARTIAL DIFFERENTIATION 5.11 Thermodynamic relations Thermodynamic relations provide a useful set of physical examples of partia ...

5.11 THERMODYNAMIC RELATIONS
Show that(∂S/∂V)T=(∂P /∂T)V. Applying (5.45) todS, with independent variablesVandT, we find dU=TdS ...

PARTIAL DIFFERENTIATION Although the Helmholtz potential has other uses, in this context it has simply provided a means for a qu ...

5.13 EXERCISES constant limits of integration the order of integration and differentiation can be reversed. In the more general ...

PARTIAL DIFFERENTIATION (a) Find all the first partial derivatives of the following functionsf(x, y): (i)x^2 y, (ii)x^2 +y^2 + 4 ...

5.13 EXERCISES 5.9 The functionf(x, y) satisfies the differential equation y ∂f ∂x +x ∂f ∂y =0. By changing to new variablesu=x^ ...

PARTIAL DIFFERENTIATION the horizontal. The cost of construction isαperunitheightofwallandβper unit (slope) length of roof. Show ...

5.13 EXERCISES O R x y θ θ 2 θ Figure 5.5 The reflecting mirror discussed in exercise 5.24. 5.26 FunctionsP(V,T),U(V,T)andS(V,T) ...

PARTIAL DIFFERENTIATION By consideringd(U−TS−HM) prove that ( ∂M ∂T ) H = ( ∂S ∂H ) T . For a particular salt, M(H, T)=M 0 [1−ex ...

5.14 HINTS AND ANSWERS 5.33 If I(α)= ∫ 1 0 xα− 1 lnx dx, α >− 1 , what is the value ofI(0)? Show that d dα xα=xαlnx, and dedu ...

PARTIAL DIFFERENTIATION 5.19 The cost always includes 2αh, which can therefore be ignored in the optimisation. With Lagrange mul ...

6 Multiple integrals For functions of several variables, just as we may consider derivatives with respect to two or more of them ...

MULTIPLE INTEGRALS V U C T S dx dy R dA=dxdy y d c a b x Figure 6.1 A simple curveCin thexy-plane, enclosing a regionR. and ∆y→0 ...

6.1 DOUBLE INTEGRALS An alternative way of evaluating the integral, however, is first to sum up the contributions from the eleme ...

MULTIPLE INTEGRALS y 1 1 dy 0 0 dx x x+y=1 R Figure 6.2 The triangular region whose sides are the axesx=0,y=0and the linex+y=1. ...