Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 1 / c

Table of contents
- Solution c

Solution

The Jacobian determinant is the oriented scalar triple product of the three coordinate basis vectors:

J = det \frac{\partial ( x , y , z )}{\partial ( u , v , ϕ )} = (h_{u} \times h_{v}) \cdot h_{ϕ} .

Here

h_{u} \times h_{v} = (u^{2} + v^{2}) (- sin ϕ, cos ϕ, 0)

, so

J = uv (u^{2} + v^{2}), d V = uv (u^{2} + v^{2}) d u d v d ϕ .

The sign is nonnegative on the stated range, and the absolute Jacobian determinant also equals

h_{u} h_{v} h_{ϕ}

as expected for orthogonal coordinates.

 Ancestors (12)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version