By Jerrold Marsden, Alan Weinstein

The 3rd of a three-volume paintings, this ebook is the outgrowth of the authors' event instructing calculus at Berkeley. It covers multivariable calculus and starts off with the required fabric from analytical geometry. It is going directly to hide partial differention, the gradient and its purposes, a number of integration, and the theorems of eco-friendly, Gauss and Stokes. The authors encourage the examine of calculus utilizing its purposes. positive factors many solved difficulties and broad routines.

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**Example text**

If r < 0 this is the decay equation; if r > 0 then it models exponential growth. This equation is separable. We write 1 dx = rdt. 3 Separable Equations 25 or ln |x| = rt + C or |x| = ert+C = eC ert . This means x = ±eC ert . Therefore, the general solution of the growth–decay equation can be written compactly as x(t) = Cert , where ±eC has been replaced by C. This is the same solution we obtained earlier in a diﬀerent way. 17 Sometimes it is convenient in an initial value problem to integrate both sides of the equation using deﬁnite integrals with variable upper limits.

11. Consider the linear diﬀerential equation x +p(t)x = q(t), and let x1 = x1 (t) and x2 = x2 (t) be solutions. (a) Show that the sum x(t) = x1 (t) + x2 (t) is a solution if, and only if, q(t) = 0. (b) If x1 = x1 (t) is a solution to x + p(t)x = 0 and x2 = x2 (t) is a solution to x + p(t)x = q(t), show that x(t) = x1 (t) + x2 (t) is a solution to x + p(t)x = q(t). 12. Show that if x1 = x1 (t) and x2 = x2 (t) are both solutions to the DE x + p(t)x = 0, then x1 /x2 is constant. Hint: Use the quotient rule for derivatives.

If the concentration of C in the reactor is given by C(t), then mass balance leads the governing equation (V C) = qCin −qC −kV C 2 . Find the constant solutions, or equilibrium states, of this equation. What are the units of k? 3 Separable Equations 35 7. (Enzyme kinetics) Work Exercise 6 if the rate of an enzyme reaction is given by Michaelis–Menten kinetics r= aC , b+C where a and b are positive constants. 8. (Batch reactor ) A batch reactor is a reactor of volume V where there are no in and out ﬂow rates.