Makanyaga
JF-Expert Member
- Sep 28, 2007
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CALCULUS: DERIVATION OF THE LAPLACE EQUATION IN SPHERICAL COORDINATES
Derivation of Laplace Equation in Spherical Coordinates is a challenging task, not that it is difficult but rather; the list of terms to be added in the process become very exhaustive to the extent that they can fill completely an entire page of an A4 sheet, whether in Portrait or Landscape orientation. In this work however, I have devised my own way of adding up such list of exhaustive terms using a method I have named as "THE TABULAR METHOD".
Under this method, each of the three transformed Cartesian components (i.e. x, y and z) has its corresponding spherical polar component, FIRST added together in its own, separate table and eventually, all the three components combined and added together in a single, FINAL table
The most important thing to note in this work is that someone perhaps new to the derivation, may somehow find it difficult to follow the procedures involved in the process, if at all s/he is not accustomed to working with 3x3 Matrices. I have therefore (right after the last page of the derivation process), included some simple practical reviews of 3x3 matrices examples, more inclined to finding inverses of 3x3 matrices
This work is part of the manual I'm currently working on, which will be made available to you (and so is to the public), immediately after its completion.
Please see more in the attachment, and let me have your feedback, if any!
Derivation of Laplace Equation in Spherical Coordinates is a challenging task, not that it is difficult but rather; the list of terms to be added in the process become very exhaustive to the extent that they can fill completely an entire page of an A4 sheet, whether in Portrait or Landscape orientation. In this work however, I have devised my own way of adding up such list of exhaustive terms using a method I have named as "THE TABULAR METHOD".
Under this method, each of the three transformed Cartesian components (i.e. x, y and z) has its corresponding spherical polar component, FIRST added together in its own, separate table and eventually, all the three components combined and added together in a single, FINAL table
The most important thing to note in this work is that someone perhaps new to the derivation, may somehow find it difficult to follow the procedures involved in the process, if at all s/he is not accustomed to working with 3x3 Matrices. I have therefore (right after the last page of the derivation process), included some simple practical reviews of 3x3 matrices examples, more inclined to finding inverses of 3x3 matrices
This work is part of the manual I'm currently working on, which will be made available to you (and so is to the public), immediately after its completion.
Please see more in the attachment, and let me have your feedback, if any!
