MTH623 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || TENSOR ANALYSIS AND ITS APLICATIONS || BY VuTech

 

MTH623 ASSIGNMENT NO. 1 FALL 2022 

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MTH623 ASSIGNMENT NO. 1 FALL 2022 || 100% RIGHT SOLUTION || TENSOR ANALYSIS AND ITS APPLICATIONS|| BY VuTech

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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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