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Calculus with Curvilinear Coordinates Problems and Solutions by Markus Antoni

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Calculus with Curvilinear Coordinates Problems and Solutions by Markus Antoni , Institute of Geodesy, University of Stuttgart, Stuttgart, Germany.
This book contains exercises and their solutions of the course ‘Advanced Mathematics’ in the GEOENGINE master program at the University of Stuttgart.
In this course, we recall and introduce the topics • ordinary differential equations of second order and system of differential equations, • vector analysis in curvilinear coordinates, including integral theorems of Green, Stokes, and Gauß, • partial differential equations and potential theory.
We present here only the vector analysis, which turned out to be a challenge for many students, due to the parametrization of objects and the different curvilinear coordinate systems. The other courses of the program deal implicitly or explicitly with several coordinate systems and coordinate transformations, e.g., • spherical coordinates: tachymetry, geodetic astronomy, gravity field representation, etc., • cylindrical coordinates: regional collocation, surveying in tunnels, TEC calculations, etc., • ellipsoidal coordinates: gravity field representation, map projection, state surveying, • modified torus coordinates: sampling and intersection of satellites tracks, • affine coordinates: misaligned observations in state surveying of nineteenth century, • triaxial ellipsoidal coordinates, prolate or oblate spheroidal coordinates: gravity field modeling of non-spherical celestial bodies.
Hence, we introduce the concept in terms of the tangential vectors of the frame. In this form, the procedures are valid for every orthogonal coordinate system, and we do not need to restrict ourself to special cases.

Calculus with Curvilinear Coordinates Problems and Solutions by Markus Antoni cover the following topics.

  • 1. Arc Length and Tangent Vector
    2. Differentiation of Field Quantities
    3. Work, Line Integral and Potential
    4. Integral Theorems of Vector Analysis

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