Mathematics · University
Jacobian of a planar transformation
An integral change of variables uses the absolute determinant; this question asks for its signed value.
Method and conditions
An integral change of variables uses the absolute determinant; this question asks for its signed value.
det ∂(u,v)/∂(x,y) = uₓvᵧ − uᵧvₓ
Worked examples
Let u = 4x² − y² and v = 5xy. Calculate the signed Jacobian determinant ∂(u,v)/∂(x,y) at (9,5). Round the final result to 4 decimal places if needed.
The derivative matrix is [[72,-10],[25,45]]. Its determinant is 2(5)[4(9)² + (5)²].
3490
Let u = 2x² − y² and v = 5xy. Calculate the signed Jacobian determinant ∂(u,v)/∂(x,y) at (4,5). Round the final result to 4 decimal places if needed.
The derivative matrix is [[16,-10],[25,20]]. Its determinant is 2(5)[2(4)² + (5)²].
570
Try it
Let u = 6x² − y² and v = 9xy. Calculate the signed Jacobian determinant ∂(u,v)/∂(x,y) at (3,5). Round the final result to 4 decimal places if needed.
Let u = 1x² − y² and v = 3xy. Calculate the signed Jacobian determinant ∂(u,v)/∂(x,y) at (3,5). Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 1422
The derivative matrix is [[36,-10],[45,27]]. Its determinant is 2(9)[6(3)² + (5)²].
- 204
The derivative matrix is [[6,-10],[15,9]]. Its determinant is 2(3)[1(3)² + (5)²].
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
AI-assisted preview; independent educational review required. Level labels describe prerequisites, not accreditation or official curriculum alignment.