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Recently I've been studying about orthogonal coordinate systems and vector operations in different coordinate systems.In my studies,I realized there are some inconsistencies between different sources which I can't resolve.
For example in Arfken,it is said that the determinant definition of the cross product is reserved after changing to a coordinate system other than Cartesian and also Foundations of the Electromagnetic theory by Reitz and Milford uses the same way for calculating a cross product in spherical coordinates as in the Cartesian system.
But in documents I found on the internet,the cross product in other coordinate systems involves the [itex] h_i(=\sqrt{g_{ii}}) [/itex] coefficients.One example is here.
But again,the calculations presented in some other documents(like this and http://math.arizona.edu/~vpiercey/Lame.pdf) are that much different in(maybe)notation that I have problem relating them.
I want to ask,is there a book which is considered as the ultimate reference that its notations and definitions are most widely used?
For example in Arfken,it is said that the determinant definition of the cross product is reserved after changing to a coordinate system other than Cartesian and also Foundations of the Electromagnetic theory by Reitz and Milford uses the same way for calculating a cross product in spherical coordinates as in the Cartesian system.
But in documents I found on the internet,the cross product in other coordinate systems involves the [itex] h_i(=\sqrt{g_{ii}}) [/itex] coefficients.One example is here.
But again,the calculations presented in some other documents(like this and http://math.arizona.edu/~vpiercey/Lame.pdf) are that much different in(maybe)notation that I have problem relating them.
I want to ask,is there a book which is considered as the ultimate reference that its notations and definitions are most widely used?