Mathematical Methods for Physics and Engineering - Matematica.NET

PARTIAL DIFFERENTIATION5.19 The cost always includes 2αh, which can therefore be ignored in the optimisation.With Lagrange multiplier λ, sinθ = λw/(4β) andβ sec θ − 1 λw tan θ = λh, leading2to the stated results.5.21 The envelope of the lines x/a + y/(c−a) − 1 = 0, as a is varied, is √ x+ √ y = √ c.Area = c 2 /6.5.23 (a) Using α =cotθ, whereθ is the initial angle a jet makes with the vertical, theequation is f(z, ρ,α) =z−ρα+[gρ 2 (1+α 2 )/(2v0 2 )], and setting ∂f/∂α = 0 givesα = v0 2/(gρ). The water bell has a parabolic profile z = v2 0 /(2g) − gρ2 /(2v0 2).(b) Setting z = 0 gives the minimum diameter as 2v0 2/g.5.25 Show that (∂G/∂P) T = V and (∂G/∂T) P = −S. From each result, obtain anexpression for ∂ 2 G/∂T∂P and equate these, giving (∂V/∂T) P = −(∂S/∂P) T .5.27 Find expressions for (∂S/∂V) T and (∂S/∂T) V , and equate ∂ 2 S/∂V∂T with5.29∂ 2 S/∂T∂V. U(V,T)=cT − aV −1 .dI/dy = −Im[ ∫ ∞0 y2 ). Integrate dI/dy from 0 to ∞.I(∞) =0andI(0) = J.5.31 Integrate the RHS of the equation by parts, before differentiating with respectto y. Repeated application of the method establishes the result for all orders ofderivative.5.33 I(0) = 0; use Leibnitz’ rule.5.35 Write x(t) =− cos t ∫ tπcos ξf(ξ) dξ and differentiate each0 tterm as a product to obtain dx/dt. Obtaind 2 x/dt 2 in a similar way. Note thatintegrals that have equal lower and upper limits have value zero. The value ofx(π) is ∫ π0186