Solution:2014/B5:Let us suppose the transformation equations
where f,g and h are continuous, have continuous partial derivatives, and have a
single-valued inverse. We now establish a one-one correspondence between
points in an xyz and rectangular coordinate system. Equation(i) can be written as
A point P can be defined by rectangular coordinates (x,y,z) and curvilinear coordinates
.
If are constant then as varies, r describes a curve which is called the
coordinate curve. Similarly we define and coordinate curves through P.
From equation (ii), we have
The vector is tangent to the coordinate curve at P. If is a unit vector at P
in this direction, we can write where .
Similarly, and where and
Therefore, equation (iii) can be written as
(a) Cylindrical Polar coordinates:
Transformation equations:
where
(b) If are mutually perpendicular at any point P, the curvillinear
coordinates are called orthogonal.
If is a scalar field, then
and
After simplification, we have
In cylindrical polar coordinates,
Therefore, from ,
we have
2014/B5 (a) Draw skecthes to illustrate R, 0 and z coordinate curves for the case of cylindrical polar coordinates (b) Show that the gradient of a scalar field, p, can be expressed in terms of cu...