Find the scale factors for U1u3 x = U3, y = U1U3 + U2 = Z U2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with the solution to the following problem. The book is very hard to follow. This is
![2. Find the scale factors for
\[ x = u_3, \quad y = u_1 u_3 + u_2, \quad z = \frac{u_1 u_3}{u_2} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1345a87f-3d4a-45ee-a13b-cb772771cc65%2F5ac68e25-fa42-41d9-9547-ac5f672bf85b%2Fcgrjhgl_processed.png&w=3840&q=75)
Transcribed Image Text:2. Find the scale factors for
\[ x = u_3, \quad y = u_1 u_3 + u_2, \quad z = \frac{u_1 u_3}{u_2} \]
![**General Orthogonal Curvilinear Coordinates**
**Scale Factors:**
\[
h_i = \left| \frac{\partial R}{\partial u_i} \right| = \frac{1}{\sqrt{u_i}} \quad (i = 1, 2, 3)
\]
**Displacement Vector:**
\[
d\mathbf{R} = h_1 du_1 \mathbf{e}_1 + h_2 du_2 \mathbf{e}_2 + h_3 du_3 \mathbf{e}_3
\]
**Arc Length:**
\[
ds = \left( h_1^2 du_1^2 + h_2^2 du_2^2 + h_3^2 du_3^2 \right)^{1/2}
\]
**Volume Element:**
\[
dV = h_1 h_2 h_3 \, du_1 \, du_2 \, du_3
\]
**Gradient:**
\[
\nabla f = \frac{1}{h_1} \frac{\partial f}{\partial u_1} \mathbf{e}_1 + \frac{1}{h_2} \frac{\partial f}{\partial u_2} \mathbf{e}_2 + \frac{1}{h_3} \frac{\partial f}{\partial u_3} \mathbf{e}_3
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1345a87f-3d4a-45ee-a13b-cb772771cc65%2F5ac68e25-fa42-41d9-9547-ac5f672bf85b%2F0er5mk_processed.png&w=3840&q=75)
Transcribed Image Text:**General Orthogonal Curvilinear Coordinates**
**Scale Factors:**
\[
h_i = \left| \frac{\partial R}{\partial u_i} \right| = \frac{1}{\sqrt{u_i}} \quad (i = 1, 2, 3)
\]
**Displacement Vector:**
\[
d\mathbf{R} = h_1 du_1 \mathbf{e}_1 + h_2 du_2 \mathbf{e}_2 + h_3 du_3 \mathbf{e}_3
\]
**Arc Length:**
\[
ds = \left( h_1^2 du_1^2 + h_2^2 du_2^2 + h_3^2 du_3^2 \right)^{1/2}
\]
**Volume Element:**
\[
dV = h_1 h_2 h_3 \, du_1 \, du_2 \, du_3
\]
**Gradient:**
\[
\nabla f = \frac{1}{h_1} \frac{\partial f}{\partial u_1} \mathbf{e}_1 + \frac{1}{h_2} \frac{\partial f}{\partial u_2} \mathbf{e}_2 + \frac{1}{h_3} \frac{\partial f}{\partial u_3} \mathbf{e}_3
\]
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