By Robert Rosen
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Those notes are according to a one-quarter path given on the division of Biophysics and Theoretical Biology of the collage of Chicago in 1916. The path was once directed to graduate scholars within the department of organic Sciences with pursuits in inhabitants biology and neurobiology. just a moderate acquaintance with chance and differential equations is needed of the reader.
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3 Although the vanishing of the first derivative is not an infallible guide to maxima and minima of a continuous functionf, it can be a very useful one. Any point Xo at which the derivative of ! 4 vanishes is called a stationary point3 • (There are, of course, tests by which one can decide whether a given stationary point is a relative maximum, a relative minimum, or an infiection point. ) Before turning to more general situations, it may be seen how these ideas are related to the discussion in the first chapter.
N). 6, it is known that a necessary condition for the minimization of 1(€l, ••• , €n) is the vanishing of the partial derivatives: 01(€1, ... , €n) 0 · ... 1 2 ~ , I , , ... , n U€j when €1 =0. 9, indicates that 01 -= O€, ft to 1( oF OF) dt 1]'-+1)1-, OU, OUt from which, after an integration by parts as before, the EulerLagrange equation is derived. oF _~ oF =0 dt oui OUt Thus, if there are n dependent variables, the problem is characterized by the solution of n equations of Euler-Lagrange type, one for each dependent variable.
A book with a similar title, stressing applications to physics and engineering, is that of R. Weinstock, McGraw-HilI, New York,1952. 40 OPTIMALITY PRINCIPLES IN BIOLOGY 8. This form for the functional J(y) emerged very early from a study of the classical problems of the calculus of variations. It seems at first sight to be of a very special nature, but in reality, such an integral representation is possible for arbitrary functionals under very general conditions. 1 Introduction THE vascular system is the name given to the system of arteries, capillaries, and veins which carries blood from the heart to all parts of the body and back again.