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MTH623 ASSIGNMENT NO. 1 FALL 2022
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Assignment # 01 MTH623 (Fall 2022)
Maximum
Marks: 20
Due
Date: 24 Nov, 2022
INSTRUCTIONS
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Question #1:
Find the Jacobian `\frac{x,y,z}{u_1,u_2,u_3}`
I) Parabolic cylindrical coordinates `x = \frac{1}{2}\left( u^2 - v^2 \right), y = uv, z = z`
where `- \infty < u < \infty , v \ge 0, - \infty < z < \infty`
II) Elliptic Cylindrical coordinates: `x = acoshu cosv, y = asinhu sinv, z = z`
Where `u \ge 0,0 \le v < 2\pi , ; , - \infty < z < \infty`
III) Prolate spheroidal coordinates:
`x = asinh\alpha sin\beta cos\gamma`
`\y = asinh\alpha sin\beta sin\gamma`
`\z = acosh\alpha cos\beta ` Where `\alpha \ge 0, 0 \le \beta \le \pi`
`0 \le \gamma < 2\pi`.
Solution:
I) Parabolic cylindrical coordinates `x = \frac{1}{2}\left( u^2 - v^2 \right), y = uv, z = z`
where `- \infty < u < \infty , v \ge 0,
- \infty < z < \infty`
II) Elliptic Cylindrical coordinates: `x = acoshu cosv, y = asinhu sinv, z = z`
Where `u \ge 0,0 \le v < 2\pi , ; , - \infty <
z < \infty`
III) Prolate spheroidal coordinates:
`x = asinh\alpha sin\beta cos\gamma`
`\y = asinh\alpha sin\beta sin\gamma`
`\z = acosh\alpha cos\beta ` Where `\alpha \ge 0, 0 \le \beta \le \pi`
`0 \le \gamma < 2\pi`.
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Question # 2:
The eight
vertices of a rectangular prism are as follows:
`\V_1 =
\left(0,0,0\right), V_2 = \left(1,0,0\right), V_3 = \left(1,2,0\right), V_4 =
\left(0,2,0\right)`
`\V_5 =
\left(0,0,3\right), V_6 = \left(1,0,3\right), V_7 = \left(1,2,3\right), V_8 =
\left(0,2,3\right)`
Find the
coordinates of the vertices after the prism is rotated counterclockwise about
the z-axis through `\theta = 60^\circ`.
Solution:
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