An Introduction to Fluid Mechanics pp 1-30 | Cite as

# Mathematical Prerequisites

## Abstract

Fluid mechanics is the mechanics of fluids embracing liquids and gases and is the discipline within a broad field of applied mechanics concerned with the behavior of liquids and gases at rest and in motion. Knowledge of ordinary and partial differential equations, linear algebra, vector calculus, and integral transforms is a fundamental prerequisite. However, to better access the underlying physical interpretations and mechanisms of fluid motions, additional mathematical knowledge is required, which is introduced in this chapter. First, the index notation with free and dummy indices is discussed, followed by the elementary theory of the Cartesian tensor, including tensor algebra and tensor calculus. Based on these, field quantities and mathematical operations which are essential to fluid mechanics in orthogonal curvilinear coordinate systems can be expressed in a coherent manner. Useful integral theorems in establishing the theory of fluid mechanics, such as Gauss’s divergence theorem, Green’s and Stokes’ theorems, are summarized as an outline. A review of complex analysis which is used intensively in discussing two-dimensional potential-flow theory of fluid mechanics is provided at the end. Detailed derivations and proofs of most equations and theorems are absent. They provide additional exercises for readers to become familiar with the topics introduced in this chapter.

## Further Reading

- R. Aris,
*Vectors, Tensors, and the Basic Equations of Fluid Mechanics*(Dover, New York, 1962)zbMATHGoogle Scholar - R.V. Churchill, J.W. Brown,
*Complex Variables and Applications*, 5th edn. (McGraw-Hill, Singapore, 1990)Google Scholar - I.G. Currie,
*Fundamental Mechanics of Fluids*, 2nd edn. (McGraw-Hill, Singapore, 1993)zbMATHGoogle Scholar - F.B. Hildebrand,
*Methods of Applied Mathematics*, 2nd edn. (Prentice-Hill, New Jersey, 1965)zbMATHGoogle Scholar - M. Itskov,
*Tensor Algebra and Tensor Calculus for Engineers: With Applications to Continuum Mechanics*(Springer, Berlin, 2015)zbMATHGoogle Scholar - J.P. Keener,
*Principles of Applied Mathematics*(Addison-Wesley, New York, 1988)zbMATHGoogle Scholar - W.M. Lai, D. Rubin, E. Krempl,
*Introduction to Continuum Mechanics*, 3rd edn. (Pergamon Press, New York, 1993)zbMATHGoogle Scholar - J.E. Marsden,
*Basic Complex Analysis*(W.H. Freeman and Company, San Francisco, 1973)zbMATHGoogle Scholar - D.E. Neuenschwander,
*Tensor Calculus for Physics*(Johns Hopkins University Press, New York, 2014)zbMATHGoogle Scholar - K.F. Riley, M.P. Hobson, S.J. Bence,
*Mathematical Methods for Physics and Engineering*(Cambridge University Press, Cambridge, 1998)zbMATHGoogle Scholar - I.S. Sokolnikoff,
*Tensor Analysis: Theory and Applications to Geometry and Mechanics of Continua*, 2nd edn. (Wiley, New York, 1969)Google Scholar - D.V. Widder,
*Advanced Calculus*, 2nd edn. (Prentice-Hill, New Jersey, 1961)zbMATHGoogle Scholar