Appendix G
Some Tools of Vector Analysis
If I were again beginning my studies, I would follow the advice
of Plato and start with mathematics.
— Galileo Galilei
G.1 Displacements and Their Length in General Coordinates
The points in a three-dimensional space can be specified by three numbers
(x 1 , x 3 , x 3 ). These might be the Cartesian coordinates of a point. But locations in
space are sometimes more conveniently specified by intersections of grid lines not
parallel to each other, such as those for spherical polar coordinates. 1 Curvilinear
coordinates are defined to have grid lines which are locally perpendicular to each
other, so that the small vector displacements can be express as
dr =
i
h i dx i e i
where the e i is a unit vector along one of the grid lines, and
e i · e j = δ ij .
Three common choices for curvilinear coordinates are Cartesian, cylindrical, and
spherical polar. In these cases, h i and the usual choices for symbols are given below:
1 The choice of coordinates is made strategically, according to the underlying symmetry of the
system. A good choice can make the solution to the functional relationships for a given problem
simpler to find and express than a bad choice.
© Springer Nature Switzerland AG 2020
W. C. Parke, Biophysics, https://doi.org/10.1007/978-3-030-44146-3
657
Some Tools of Vector Analysis
If I were again beginning my studies, I would follow the advice
of Plato and start with mathematics.
— Galileo Galilei
G.1 Displacements and Their Length in General Coordinates
The points in a three-dimensional space can be specified by three numbers
(x 1 , x 3 , x 3 ). These might be the Cartesian coordinates of a point. But locations in
space are sometimes more conveniently specified by intersections of grid lines not
parallel to each other, such as those for spherical polar coordinates. 1 Curvilinear
coordinates are defined to have grid lines which are locally perpendicular to each
other, so that the small vector displacements can be express as
dr =
i
h i dx i e i
where the e i is a unit vector along one of the grid lines, and
e i · e j = δ ij .
Three common choices for curvilinear coordinates are Cartesian, cylindrical, and
spherical polar. In these cases, h i and the usual choices for symbols are given below:
1 The choice of coordinates is made strategically, according to the underlying symmetry of the
system. A good choice can make the solution to the functional relationships for a given problem
simpler to find and express than a bad choice.
© Springer Nature Switzerland AG 2020
W. C. Parke, Biophysics, https://doi.org/10.1007/978-3-030-44146-3
657