User:EGM6321.f12.team7.Marjanovic/report5/5.14

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Problem 6: Finding the Laplacian in Spherical Coordinates using the Math/Physics Convention
Based on lecture notes [ http://upload.wikimedia.org/wikiversity/en/3/39/Pea1.f12.sec40.djvu Section 40]

Given: The Laplace Operator In General Curvilinear Coordinates
The Laplacian in general curvilinear coordinates is given as:

Also, the infinitesimal length ds is given as:

Lastly, the Math/Physics convention for spherical coordinates is defined as:

Where

$$\displaystyle \bar\theta = \frac{\pi}{2} - \theta $$

Find: The Laplacian in Spherical Coordinates
Find the Laplacian of u in spherical coordinates using the math/physics convention.

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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In order to obtain $$\displaystyle h_{1}, h_{2}, h_{3} $$ we must make the proper substitutions in ($$). We know from the cofunction identities in trigonometry that:

Furthermore,

($$) then becomes:

Here we see that

Employing this in ($$), we get:

This can be further simplified to arrive at the final solution: