sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition
sin 8₂ = ਸ 1²+1 Show that at this stationary value f(r) has a minimum. sin 8₁ = Exercise 1.21 Show that the functionals La S₁[y] = dr (1+ry') y' and S₂[y] = d-r √(d-x)² + h where b> a>0, y(b) = B and y(a) = A, are both stationary on the same curve, namely In(x/a) In(b/a) y(x) = A + (B-A): Explain why the da r 38 Solution 1.21 Observe that -Sa Si[y] = S₂[y] + how ?? Come dry (r) = S₂[y]+B-A, 8 = 2€ = 26 [* dx xy' (a)g'(x) + 0(e²) Chap that is, the values of the two functionals differ by a constant, independent of the path. Hence the stationary paths of the two functionals are the same. Consider the difference & Saly+eg] - S2[y], where g(a) = g(b) = 0: so that 8 = O(²) if ry'(x) = e, where e is a constant. Integrating this equation gives y(x) = d+ cln(x/a), where d is another constant. The boundary condition
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.3: Implicit Differentiation
Problem 45E
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