Abstract
The balances of continuum theory as well as the constitutive equations contain spatial derivatives of scalar vector and of tensor fields. In this chapter we will learn how to express these in arbitrary curvilinear coordinates. This means that we will now truly switch from tensor algebra to tensor analysis. In index notation spatial derivatives will be handled by means of the covariant derivative and Christoffel symbols. In absolute notation we will introduce the so-called del operator. Both will be specified for technically relevant coordinate systems, namely cylindrical, spherical, and elliptical coordinates. The formalism will be applied to balance and constitutive equations in subsequent chapters.
In the fall of 1972 President Nixon announced that the
rate of increase of inflation was decreasing.
This was the first time a sitting president used
the third derivative to advance his case for reelection.
Hugo Rossi, Professor Emeritus of Mathematics, University of Utah
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Notes
- 1.
The tildes and circumflexes do not refer to the Lagrangian and Eulerian ways of descriptions from Sect. 3.4. They are merely a hint that different functions were used depending upon the choice of coordinates.
- 2.
Note that the symbol f for the scalar field does not contain any information about the coordinate system that was used. It is to be understood in an absolute sense just like the symbol A denotes an absolute vector. The contravariant components of the gradient in Eq. (4.1.3) can be obtained by multiplication with the metric g lk.
- 3.
In absolute notation it is also customary to write grad f instead of ∇f.
- 4.
or div grad \( \left( \bullet \right) \).
- 5.
The issue how the volume element transforms during reflections is not addressed in this section. See Sect. 8.4 for more details.
References
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Müller, W.H. (2014). Spatial Derivatives of Fields. In: An Expedition to Continuum Theory. Solid Mechanics and Its Applications, vol 210. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7799-6_4
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