Fundamental equation differential calculus

We reached this point from the discussion just prior to equation 44-64, and there we noted that a reader of the original column felt that equation 44-64 was being incorrectly used. Equation 44-64, of course, is a fundamental equation of elementary calculus and is itself correct. The problem pointed out was that the use of the derivative terms in equation 44-64 implicitly states that we are using the small-noise model, which, especially when changing the differentials to finite differences in equation 44-65, results in incorrect equations. 

Notes on Three-dimensional Differential Calculus and the Fundamental Equations of Electrostatics... 

Our problem is to take the uncertainties in x, X2, and so on, and calculate the uncertainty in y, the dependent variable. This is called the propagation of errors. If the errors are not too large, we can take an approach based on a differential calculus. The fundamental equation of differential calculus is Eq. (7.9),... 

After rearrangement to the proper differential format, invoke the fundamental lemma of calculus to produce a differential equation. 

We need to calculate the expected error in y from the expected errors in x, X2, , Xn- We take an approach based on the fundamental equation of differential calculus, Eq. (8.12)... 

We will discuss quantum mechanics extensively in Chapters 5 and 6. It provides the best description we have to date of the behavior of atoms and molecules. The Schrodinger equation, which is the fundamental defining equation of quantum mechanics (it is as central to quantum mechanics as Newton s laws are to the motions of particles), is a differential equation that involves a second derivative. In fact, while Newton s laws can be understood in some simple limits without calculus (for example, if a particle starts atx = 0 and moves with constant velocity vx,x = vxt at later times), it is very difficult to use quantum mechanics in any quantitative way without using derivatives. 

It is presumed that the reader who desires a complete understanding of the contents of this book has had at least a course in advanced calculus and preferably a general course also in partial differential equations and boundary-value problems, or a first course in the methods of mathematical physics. It is also assumed that he is acquainted with the fundamental concepts involved in modern physics and has been introduced to... 

Den Hartog, J. P. 1985. Mechanical Vibrations, 4th ed. New York Dover Publications. Reprint. Originally published 1956. New York McGraw-Hill. Covers the fundamentals of mechanical vibrations. Can be used by practicing engineers as well as for classroom instruction. An elementary knowledge of dynamics and calculus is necessary, but differential equations are explained in detail. Examples have been drawn from real-life experiences of the author and his friends. Each chapter includes problems that illustrate typical practical situations with answers included at the back. 

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