Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 1 / 17C / a

Table of contents
- Solution a

Solution

Let

ω = \nabla \times u

be the vorticity. In component notation, contracting two Levi-Civita symbols gives

(u \times ω)_{i} = ε_{ijk} u_{j} ε_{k l m} \partial_{l} u_{m} = u_{j} \partial_{i} u_{j} - u_{j} \partial_{j} u_{i} .

The first term is

\partial_{i} (∣ u ∣^{2} /2)

and the second is the

i

th component of the convective acceleration. Rearranging proves the velocity identity

(u \cdot \nabla) u = \nabla (\frac{1}{2} ∣ u ∣^{2}) - u \times ω .

 Ancestors (10)

 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