Two-dimensional parabolic coordinates are defined in the -plane by x = uv

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Curvilinear and orthogonal coordinates

Problem

Two-dimensional parabolic coordinates

(u,v)

are defined in the

xy

-plane by

x = uv, \qquad y = \tfrac{1}{2}(v^2-u^2),

with

u > 0

and

v

any real number, so that no two pairs

(u, v)

label the same point. ((a)) Write the position vector

\mathbf r(u,v) = x\hat{\mathbf x} + y\hat{\mathbf y}

, find

\partial\mathbf r/\partial u

and

\partial\mathbf r/\partial v

, and hence obtain the scale factors

h_u

and

h_v

. ((b)) Show that the system is orthogonal by evaluating

\partial\mathbf r/\partial u \cdot \partial\mathbf r/\partial v

. ((c)) Given that

x

and

y

are lengths, state the dimensions of

u

and

v

, and check that

h_u\,\mathrm du

still comes out with the dimensions of a length.

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