Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5 Partial differentiation In chapter 2, we discussed functionsfof only one variablex, which were usually writtenf(x). Certain co ...

PARTIAL DIFFERENTIATION toxandyrespectively, and they are extremely important in a wide range of physical applications. For a fu ...

5.2 THE TOTAL DIFFERENTIAL AND TOTAL DERIVATIVE Only three of the second derivatives are independent since the relation ∂^2 f ∂x ...

PARTIAL DIFFERENTIATION can be obtained. It will be noticed that the first bracket in (5.3) actually approxi- mates to∂f(x, y+∆y ...

5.3 EXACT AND INEXACT DIFFERENTIALS this partial derivative account must be taken only ofexplicitappearances ofx 1 in the functi ...

PARTIAL DIFFERENTIATION it exact. Consider the general differential containing two variables, df=A(x, y)dx+B(x, y)dy. We see tha ...

5.4 USEFUL THEOREMS OF PARTIAL DIFFERENTIATION 5.4 Useful theorems of partial differentiation So far our discussion has centred ...

PARTIAL DIFFERENTIATION From equation (5.5) the total differential off(x, y) is given by df= ∂f ∂x dx+ ∂f ∂y dy, but we now note ...

5.6 CHANGE OF VARIABLES x y ρ φ Figure 5.1 The relationship betweenCartesian and planepolar coordinates. For each different valu ...

PARTIAL DIFFERENTIATION Thus, from (5.17), we may write ∂ ∂x =cosφ ∂ ∂ρ − sinφ ρ ∂ ∂φ , ∂ ∂y =sinφ ∂ ∂ρ + cosφ ρ ∂ ∂φ . Now it i ...

5.7 TAYLOR’S THEOREM FOR MANY-VARIABLE FUNCTIONS
Find the Taylor expansion, up to quadratic terms inx− 2 andy− 3 ,off(x, y)=yex ...

PARTIAL DIFFERENTIATION theorem then becomes f(x)=f(x 0 )+ ∑ i ∂f ∂xi ∆xi+ 1 2! ∑ i ∑ j ∂^2 f ∂xi∂xj ∆xi∆xj+···, (5.20) where ∆x ...

5.8 STATIONARY VALUES OF MANY-VARIABLE FUNCTIONS P S y x B Figure 5.2 Stationary points of a function of two variables. A minimu ...

PARTIAL DIFFERENTIATION To establish just what constitutes sufficient conditions we first note that, since fis a function of two ...

5.8 STATIONARY VALUES OF MANY-VARIABLE FUNCTIONS
Show that the functionf(x, y)=x^3 exp(−x^2 −y^2 )has a maximum at the point( √ ...

PARTIAL DIFFERENTIATION minimum x y − (^2) − (^1123) 2 − 3 −^2 − 0. 2 − 0. 4 0. 2 0. 4 0 0 0 maximum Figure 5.3 The functionf(x, ...

5.9 STATIONARY VALUES UNDER CONSTRAINTS where thearare coefficients dependent upon ∆x. Substituting this into (5.26), we find ∆f ...

PARTIAL DIFFERENTIATION varied. However, it is often the case in physical problems that not all the vari- ables used to describe ...

5.9 STATIONARY VALUES UNDER CONSTRAINTS
The temperature of a point(x, y)on a unit circle is given byT(x, y)=1+xy.Findthe temper ...

PARTIAL DIFFERENTIATION
Find the stationary points off(x, y, z)=x^3 +y^3 +z^3 subject to the following constraints: (i)g(x, y, ...