Orthogonal Curvilinear coordinates [and more] for Engineers and Physical Scientists
The theory of orthogonal curvilinear coordinates is presented through a simple matrix notation avoiding the standard tensor sub-index symbolism perhaps less familiar to engineers and physical scientists.
Chapters
01
Chapter 01
Basics about reference coordinate systems and curvilinear coordinates. Rotation matrices and the transformation between global frames. Definition of local frames. Angular velocity matrices. Transformation of variance-covariance matrices of coordinates referred to local frames. Variance-covariance matrices of transformed coordinates and velocities. Computation of the 14 parameters of time-variant similarity transformations.
02
Chapter 02
Definition of the Jacobian, metric, Lamé and rotation matrices. Partial derivatives of these matrices and their transposes and inverses. Smith’s differential transformation. Alternative derivation of some equations published in 1840 by Gabriel Lamé.
03
Chapter 03
Differential matrix operators. Gradient, divergence, curl. Higher order total differentials. The transpose of a tensor-matrix of third rank. Tensors of inertia in Cartesian coordinates. Associate tensor of inertia of second-rank. Gravitational potential. First terms of the expansion of the gravitational attraction as a function of the inertia tensors. Variance-covariance matrix of the eigenvalues and eigenvectors of a rank-two symmetric tensor.
04
Chapter 04
The Christoffel symbols. Christoffel symbols for orthogonal curvilinear coordinates. Christoffel symbols of third kind. Transformation of Christoffel symbols.
05
Chapter 5
Analyses of curvilinear coordinates. Velocity and acceleration in curvilinear coordinates. Gradient of a scalar field U in curvilinear coordinates. Laplacian of a scalar function U. The divergence of a vector field in orthogonal curvilinear coordinates. The curl of a vector in orthogonal curvilinear coordinates. The dilatation gradient vector. The deviatoric rotation gradient tensor. The deviatoric stretch gradient tensor.
06
Chapter 6
Miscellaneous applications. Strain tensor. Strain tensor in orthogonal curvilinear coordinates. Second derivatives of the deviatoric strain tensor. Second order terms of the strain tensor. The deformation rotation tensor. Symmetric rotation gradient tensor. Navier-Stokes equations.
A1
Chapter A1
Applications to spherical coordinates.
A2
Chapter A2
Applications to curvilinear geodetic coordinates.
A3
Chapter A3
Applications to ellipsoidal coordinates
A4
Chapter A4
Elliptic coordinates on a triaxial ellipsoid
Book Abstract
Orthogonal Curvilinear coordinates [and more] for Engineers and Physical Scientists
The theory of orthogonal curvilinear coordinates is presented through a simple matrix notation avoiding the standard tensor sub-index symbolism perhaps less familiar to engineers and physical scientists. This could facilitate the comprehension of the material to most graduated and/or advanced students trying to grasp this somewhat intricate subject.
B-0-Front-Cover
B-1-Gradient
B-2-Smith's-Eq
B-3-Christoffel
B-4-Divergence
B-6-Second-PD
B-7-Sph-Coord
B-8-Geod-Coord
B-9-Ellip-Coord
B-5-CurlCurl
