Titlebook: Mathematical Physics; Applications and Pro V. Balakrishnan Textbook 2020 Authors 2020 Vector calculus and applications.Linear vector spaces

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Fourier Series,l physics: the latter event is often identified with the publication of Joseph Fourier’s treatise, . (.) in 1822. Fourier series and transforms have been very extensively studied and vastly generalized since then. The part of modern mathematics concerned with this subject is called ..

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Discrete Probability Distributions, random variables and the theory of probability. The subject is a vast one, and several courses are needed to do justice to it. But my objective here is far more modest:  to familiarize you with the elementary aspects of the subject that are most relevant to physical applications.

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,Gaussian Integrals, Stirling’s Formula, and Some Integrals,The function ., called a Gaussian, appears everywhere in the mathematical sciences. It plays a fundamental role in probability and statistics. We will discuss this aspect further in Chap. 20, Sect. ..

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Some More Functions,The functions we come across in physical problems may not always occur in a completely explicit form. Quite frequently, a function is specified by means of an integral. This is termed an . of the function concerned.

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Vectors and Tensors,At school, we learn that a vector is a quantity with a magnitude and a direction—in contrast to a scalar, with which no direction is associated. We then proceed to physical examples of vectors such as velocity and force, which help us understand “intuitively” how to handle vectors.

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Some More Vector Calculus,From the point of view of physical applications, the most important among the theorems of vector calculus are . and ..

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Linear Vector Spaces,Linear vector spaces comprise a basic topic in mathematics, besides occurring in many forms in a very large variety of physical applications. Foremost among these is quantum mechanics, for which linear vector spaces provide the natural language. It is therefore helpful to use Dirac notation right from the beginning.

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More About Matrices,As I have mentioned already, we may also view matrices as the representations of . acting on the elements of an LVS. In an .-dimensional Euclidean space (denoted by .), the natural basis  is given by the ket vectors represented by the column matrices

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