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Quantum Optics for Engineers by F.J. Duarte

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Appendix G: Calculus Basics

G.1  Calculus Basics

Here, we provide a brief and pragmatic survey of some useful well-known calculus mechanics and rules. A good reference on calculus is Flanders et al. (1970).

G.1.1  Differentiation Product Rule

d(fg)dx=fdgdx+gdfdx

(G.1)

Example: Differentiate the product xeikx:

d(xeikx)dx=x(ik)eikx+eikx=eikx(1+ikx)

G.1.2  Differentiation Quotient Rule

d(f/g)dx=g2(gdfdxfdgdx)

(G.2)

G.1.3  Differentiation Power Rule

If n is an integer,

d(fn)dx=nfn1dfdx

(G.3)

Example: Differentiate (x2 + 1)2:

d(x2+1)2dx=4x(x2+1)

G.1.4  Differentiation Chain Rule

If y and x are functions of t,

dydt=dydxdxdt

(G.4)

Example: Differentiate the function y=et2+2t+1 Set y = ex and x = t2 + 2t + 1. Then apply the chain ...

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