I'm coming across several things a week in my homework and classes that are suddenly crystal clear. These are topics that, had I gone to school in a more linear fashion, should have been clear to me three or four years ago, but now will do. Here's the first of them, a better interpretation of matrix multiplies with column vectors.
The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!
What do we actually want?
To convert the Cartesian nabla
to the nabla for another coordinate system, say… cylindrical coordinates.
What we’ll need:
1. The Cartesian Nabla:
2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:
3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:
How to do it:
Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.
The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…
What do we actually want?
To convert the Cartesian nabla
to the nabla for another coordinate system, say… cylindrical coordinates.
What we’ll need:
1. The Cartesian Nabla:
2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:
3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:
How to do it:
Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.
The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…
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