# Derivatives and Integrals

• Wilhelm Flügge

## Abstract

We have differentiated vector components on page 25 and have introduced the comma notation for the derivatives in definition (2.12). In rectilinear coordinates the changes of the vector components indicate the change of the vector. This is no longer so when the coordinates are curvilinear. As a simple example consider Figure 5.1 a. It shows a polar coordinate system and in it three vectors v. The vectors are equal, but their radial components v1 are not and neither are the circumferential components v2. On the other hand, the vectors in Figure 5.1b are different, but they have the same radial component v1 and the same circumferential component v2 = 0.

Anisotropy

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### References

1. [20]
E. Madelung, Die mathematischen Hilfsmittel des Physikers (7th ed.) ( Berlin: Springer, 1964 ).Google Scholar
2. [3]
L. Brillouin, Les Tenseurs en Mécanique et en Elasticité ( Paris: Masson, 1946 ).
3. [31]
J. L. Synge and A. Schild, Tensor Calculus ( Toronto: Univ. of Toronto Press, 1949 ).
4. [1]
R. Arts, Vectors, Tensors, and the Basic Equations of Fluid Mechanics ( Englewood Cliffs, N.J.: Prentice-Hall, 1962 ).Google Scholar
5. [6]
C. Eringen, Mechanics of Continua ( New York: Wiley, 1967 ).
6. [2]
H. D. Block, Introduction to Tensor Analysis (Columbus, Ohio: Merrill, 1962 ).Google Scholar
7. [30]
I. S. Sokolnikoff, Tensor Analysis (2nd ed.) ( New York: Wiley, 1964 ).
8. [14]
G. A. Hawkins, Multilinear Analysis for Students in Engineering and Science ( New York: Wiley, 1963 ).Google Scholar
9. [18]
T. Levi-Civrra, The Absolute Differential Calculus ( New York: Hafner, 1926 ).Google Scholar
10. [35]
A. P. Wills, Vector Analysis ( New York: Prentice-Hall, 1931 ).
11. [5]
A. Duscxek and A. Hochrainer, Grundzüge der Tensorrechnung in analytischer Darstellung (Wien: Springer, 1946, 1961, 1955 ), Vol. 1, 2, 3.Google Scholar
12. [4]
N. Coburn, Vector and Tensor Analysis ( New York: Macmillan, 1955 ).
13. [13]
A. E. Green and W. Zerna, Theoretical Elasticity ( Oxford: Clarendon, 1954 ).
14. [17]
H. Lass, Elements of Pure and Applied Mathematics ( New York: McGraw-Hill, 1957 ).
15. [28]
A. Schild, “Tensor Analysis,” in W. Flügge (ed.), Handbook of Engineering Mechanics ( New York: McGraw-Hill, 1962 ).Google Scholar