How do I Find the Jacobi Matrix and Its Determinant for a Given Transformation?

[Kork]

Homework Statement

The transformation f is defined by: R^2 --> R^2 and is defined by:
f(x,y) = (y^5, x^3)

Find the jacobi matrix and its determinant

Homework Equations

f(x,y) = (y^5, x^3)

The Attempt at a Solution

I would start by differentiating y^5 with respect to x and then y, then differentiate x^3 with respect to x and then y.

I end up with:

Df = (0... 5y^4 )
...(3x^2... 0 )

And from here I don't know what to do. Usually I would be told, that Df = (2,1) for example, and then I would place them instead of y and x, but here I am not given any other information than what I have written above. How do I find the determinant?

 

[HallsofIvy]

How about just doing what you were told to do? Yes, if you were asked to find the Jacobian matrix and its determinant at, say, (1, 2) you would replace x and y with those. But you aren't asked to do that so just leave them as "x" and "y".

 

[Kork]

How about just doing what you were told to do? Yes, if you were asked to find the Jacobian matrix and its determinant at, say, (1, 2) you would replace x and y with those. But you aren't asked to do that so just leave them as "x" and "y".

Okay, thank you.

But I have another question, what is the difference between the Jacobi Matrix and the Jacobi determinant?

 

[Kork]

I am also asked to:

let D be the set of points (x,y) in R^2 doe which 0 ≤ x ≤ 1 and

0 ≤ y ≤ 1. Find a function g: R^2 --> R for which

∫¹₀∫¹₀ h(x,y)dxdy =
∫¹₀∫¹₀ h(y⁵, x³*g(x,y)dxdy

is true for all functions h: D --> R integrable over D