Mathematical Methods for Physics and Engineering : A Comprehensive Guide

9.4 EXERCISES the figure and obtain three linear equations governing the currentsI 1 ,I 2 andI 3. Show that the only possible fr ...

NORMAL MODES 9.8 (It is recommended that the reader does not attempt this question until exercise 9.6 has been studied.) Find a ...

9.5 HINTS AND ANSWERS label2 1 2 3 m M m 2 km kM kM (a)ω^2 =c+ d m (c)ω^2 =c+ 2 d M + d m Figure 9.6 The normal modes, as viewed ...

10 Vector calculus In chapter 7 we discussed the algebra of vectors, and in chapter 8 we considered how to transform one vector ...

10.1 DIFFERENTIATION OF VECTORS a(u) a(u+∆u) ∆a=a(u+∆u)−a(u) Figure 10.1 A small change in a vectora(u) resulting from a small c ...

VECTOR CALCULUS y x φ ˆeφ eˆρ ρ i j Figure 10.2 Unit basis vectors for two-dimensional Cartesian and plane polar coordinates. Th ...

10.1 DIFFERENTIATION OF VECTORS in terms ofiandj. From figure 10.2, we see that ˆeρ=cosφi+sinφj, eˆφ=−sinφi+cosφj. Sinceiandjare ...

VECTOR CALCULUS The order of the factors in the terms on the RHS of (10.6) is, of course, just as important as it is in the orig ...

10.2 INTEGRATION OF VECTORS Note that the differential of a vector is also a vector. As an example, the infinitesimal change in ...

VECTOR CALCULUS x z y C O ˆb ˆt nˆ P r(u) Figure 10.3 The unit tangentˆt, normalˆnand binormalbˆto the space curveC at a particu ...

10.3 SPACE CURVES This parametric representation can be very useful, particularly in mechanics when the parameter may be the tim ...

VECTOR CALCULUS Therefore, remembering thatu=x, from (10.12) the arc length betweenx=aandx=b is given by s= ∫b a √ dr du · dr du ...

10.3 SPACE CURVES so we finally obtain dˆb ds =−τˆn. (10.14) Taking the dot product of each side withnˆ, we see that the torsion ...

VECTOR CALCULUS Finally, we note that a curver(u) representing the trajectory of a particle may sometimes be given in terms of s ...

10.5 SURFACES x y z S O T P r(u, v) ∂r ∂u ∂r ∂v v=c 2 u=c 1 Figure 10.4 The tangent planeTto a surfaceSat a particular pointP; u ...

VECTOR CALCULUS total derivative, the tangent to the curver(λ) at any point is given by dr dλ = ∂r ∂u du dλ + ∂r ∂v dv dλ . (10. ...

10.6 SCALAR AND VECTOR FIELDS A normalnto the surface at this point is then given by n= ∂r ∂θ × ∂r ∂φ = ∣∣ ∣ ∣ ∣∣ ∣ ijk acosθcos ...

VECTOR CALCULUS mathematical point of view, as we do below. In the following chapter, however, we will discuss their geometrical ...

10.7 VECTOR OPERATORS φ=constant ∇φ a P Q dφ ds in the directiona θ Figure 10.5 Geometrical properties of∇φ.PQgives the value of ...

VECTOR CALCULUS
For the functionφ=x^2 y+yzat the point(1, 2 ,−1), find its rate of change with distance in the directiona=i+2j+ ...