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Problem 3: Derive infinitesimal length ds in spherical coord
Report problem 7.3 from.

Given
1. Infinitesimal length ds is given as

in Cartesian coord.

Cartesian Coord. in terms of spherical coord.:

$$h_i$$= magnituge of the tangent vector $$mathbf g_i$$ to the coordinate line $$\xi_i$$

2. Laplace operator in general curvilinear coord. is

in Spherical coord:

at i=1,

Find
1. Show that in spherical coord.

where

2. Laplace operator at i=2 and i=3 and the final form in spherical coordinates.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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1. From ($$), ($$) and ($$)

From ($$),

where,

2. at i=2,

at i=3,

Thus, the Laplace operator in spherical coordinate is