Find the Jacobian `\frac{x,y,z}{u_1,u_2,u_3}` | Query Point Official

Find the Jacobian `\frac{x,y,z}{u_1,u_2,u_3}`

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:

Coordinate Systems and Jacobian Transformation

I) Parabolic Cylindrical Coordinates

`x = \frac{1}{2}(u^2 - v^2), y = uv, z = z`
Where `-∞ < u < ∞, v ≥ 0, -∞ < z < ∞`

II) Elliptic Cylindrical Coordinates

`x = a cosh(u) cos(v), y = a sinh(u) sin(v), z = z`
Where `u ≥ 0, 0 ≤ v < 2π, -∞ < z < ∞`

III) Prolate Spheroidal Coordinates

`x = a sinh(α) sin(β) cos(γ)`
`y = a sinh(α) sin(β) sin(γ)`
`z = a cosh(α) cos(β)`
Where `α ≥ 0, 0 ≤ β ≤ π, 0 ≤ γ < 2π`

Summary

• Jacobian is key in transforming volume elements between coordinate systems.
• Always construct the partial derivatives matrix correctly.
• Use absolute value of determinant for volume scaling in integrals.

FAQ

What is a Jacobian?

The determinant of the matrix of first-order partial derivatives; describes how a transformation scales areas or volumes.

When to use it?

In multiple integrals, coordinate transformations, and physics problems involving change of variables.

Can I calculate Jacobian in 3D coordinates?

Yes, by using `(x, y, z)` with respect to `(u, v, w)` or other transformed variables.

Related Topics

See Mathematics Notes & MCQs for more problems on calculus and multiple integrals.