Derivatives and Integrals
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.
KeywordsVector Field Covariant Derivative Area Element Polar Coordinate System Tensor Form
Unable to display preview. Download preview PDF.
- E. Madelung, Die mathematischen Hilfsmittel des Physikers (7th ed.) ( Berlin: Springer, 1964 ).Google Scholar
- R. Arts, Vectors, Tensors, and the Basic Equations of Fluid Mechanics ( Englewood Cliffs, N.J.: Prentice-Hall, 1962 ).Google Scholar
- H. D. Block, Introduction to Tensor Analysis (Columbus, Ohio: Merrill, 1962 ).Google Scholar
- G. A. Hawkins, Multilinear Analysis for Students in Engineering and Science ( New York: Wiley, 1963 ).Google Scholar
- T. Levi-Civrra, The Absolute Differential Calculus ( New York: Hafner, 1926 ).Google Scholar
- A. Duscxek and A. Hochrainer, Grundzüge der Tensorrechnung in analytischer Darstellung (Wien: Springer, 1946, 1961, 1955 ), Vol. 1, 2, 3.Google Scholar
- A. Schild, “Tensor Analysis,” in W. Flügge (ed.), Handbook of Engineering Mechanics ( New York: McGraw-Hill, 1962 ).Google Scholar