Gradient, divergence and curl in curvilinear coordinates

In the first lecture of the second part of this course we move more to consider properties of fields. We introduce three field operators which reveal interesting collective field properties, viz. • the gradient of a scalar field, • the divergence of a vector field, and • the curl of a vector field. There are two points to get over about each: • The mechanics of taking the grad, div or curl, for which you will need to brush up your multivariate calculus. • The underlying physical meaning — that is, why they are worth bothering about. In Lecture 6 we will look at combining these vector operators. 5.1 The gradient of a scalar field Recall the discussion of temperature distribution throughout a room in the overview, where we wondered how a scalar would vary as we moved off in an arbitrary direction. Here we find out how. If U(r) = U(x, y , z) is a scalar field, ie a scalar function of position r = [x, y , z] in 3 dimensions, then its gradient at any point is defined in Cartesian coordinates by gradU = ∂U ∂xˆı + ∂U ∂yˆ + ∂U ∂zˆk. (5.1)

more about vectors in cylindrical and spherical co-ordinates

Journal of Electromagnetic Analysis and Applications, 2012

A great number of semi-analytical models, notably the representation of electromagnetic fields by integral equations are based on the second order vector potential (SOVP) formalism which introduces two scalar potentials in order to obtain analytical expressions of the electromagnetic fields from the two potentials. However, the scalar decomposition is often known for canonical coordinate systems. This paper aims in introducing a specific SOVP formulation dedicated to arbitrary non-orthogonal curvilinear coordinates systems. The electromagnetic field representation which is derived in this paper constitutes the key stone for the development of semi-analytical models for solving some eddy currents modelling problems and electromagnetic radiation problems considering at least two homogeneous media separated by a rough interface. This SOVP formulation is derived from the tensor formalism and Maxwell's equations written in a non-orthogonal coordinates system adapted to a surface characterized by a 2D arbitrary aperiodic profile.

Equilibrium equations and boundary conditions of the strain gradient theory in arbitrary curvilinear coordinates have been obtained. Their special form for an axisymmetric plane strain problem is also given.

2023

It contains various problems and solutions on vector calculus

Geophysical Journal International, 2012

We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3-D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases (potential over surface or annulus, and field over surface) orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic 'power spectra', and geodetic 'degree variance', as these no longer correspond exactly to the mean-square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

2022

The Maxwell equations for the spherical components of the electromagnetic fields outside sources do not separate into equations for each component alone. We show, however, that general solutions can be obtained by separation of variables in the case of azimuthal symmetry. Boundary conditions are easier to apply to these solutions, and their forms highlight the similarities and differences between the electric and magnetic cases in both time-independent and time-dependent situations. Instructive examples of direct calculation of electric and magnetic fields from localized charge and current distributions are presented.

Journal of Informatics and Mathematical Sciences, 2018

Many real world problems are governed by non-linear differential equations. These may be single or system of ordinary or partial differential equations. In practice, problems involving single differential equations are mostly solved using analytical procedures, while those with systems of equations are solved numerically. A semi-analytical procedure namely Differential Transform Method (DTM) obtained from Taylor series in Cartesian coordinates is being used to solve linear or nonlinear equations in practice. This paper introduces the Taylor series and DTM for general orthogonal curvilinear coordinates and focuses mainly on DTM in standard two-dimensional polar coordinates and three-dimensional cylindrical polar and spherical polar coordinates.