User:Egm6321.f12.team7.parikh.a/R7.5

=R*7.5=

Problem: Heat conduction on a cylinder
Problem R*7.5 from lecture notes section 40.

Given: A generalized coordinate system
Given the coordinate transformation:

Find: Various quantities

 * 1) Find $$\{dx_i \} = \{dx_1,dx_2,dx_3\}$$ in terms of $$\{ \xi _{j} \}= \{ \xi _{1},\xi _{2},\xi _{3}\}$$ and $$\{ d\xi _{j} \}= \{ d\xi _{1},d\xi _{2},d\xi _{3}\}$$.


 * 1) Find $$ds^2 = \sum_{i}(dx_i)^2 = \sum_{k} (h_k)^2(d\xi_k)^2$$ and identify $$\{h_i\}$$ in terms of $$\{\xi_i\}$$.


 * 1) Find $$\Delta u$$ in cylindrical coordinates.


 * 1) Use separation of variables to find the separated equations and compare to the Bessel equation, given by:

Solution: Find the quantities
First, use the chain rule

Substituting in the coordinate transformations given by ($$) through ($$)

Now, plugging into the expression for $$ds^2$$ and simplifying with trigonometric identities and cancelations

Therefore,

The Laplace operator in general curvilinear coordinates is given by

Substituting yields

Now assume the solution is $$u = R(\xi _{1})\Theta(\xi _{2})Z(\xi _{3})$$, take the Laplacian and divide by $$R \Theta Z$$

Since the last term is the only term with $$\xi_3$$, and does not depend on the other curvilinear coordinates, it must independently be zero.

Multiplying the rest of the equation by $$\xi_{1}^{2}$$ again causes the first 2 terms to only depend on $$\xi_1$$ and the last term to only depend on $$\xi_2$$, therefore they are independently zero. Rearranging and simplifying yields the Bessel equation,

where