k-Jet of a function $f:R^3 \rightarrow R$

Question

I am studying this article. Can someone explain to me why the 4-jet of the function

$U(x, y, z) = x^4 + y^4 - z^6$

is equal to

$(j^{(4)} U)(x, y, z) = x^4 + y^4$

Also, the author says that the origin is an isolated critical point of the potential U, and the origin is not a minimum point.

In order to show that it is not a minimum point, I calculated the Hessian, which is the null matrix. How to show that the critical point is isolated? I searched online and found something about Morse's Lemma, is it the right way?

Thanks in advance.

Answer 1

The function given is $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $f(x, y, z) = x^4 + y^4 - z^6$.

1. 4-jet of the function at the origin

The 4-jet of the function at the origin is the 4th order Taylor series expansion of the function at the origin. The 4-jet of $f$ at the origin is $x^4 + y^4$, which is the part of the function up to 4th degree terms.

2. Critical points of the function

The critical points of the function are the points where the gradient of the function is zero. The gradient of $f$ is given by $\nabla f = (4x^3, 4y^3, -6z^5)$. Setting this to zero gives the critical points. The only critical point of $f$ is at the origin (0, 0, 0).

3. Check if the origin is an isolated critical point

The origin is an isolated critical point if there are no other critical points in its immediate vicinity. Since the only critical point of $f$ is at the origin, the origin is an isolated critical point.

4. Hessian matrix at the origin

The Hessian matrix of a function is the matrix of its second partial derivatives. The Hessian matrix of $f$ at the origin is a zero matrix, as all second partial derivatives of $f$ at the origin are zero.

5. Check if the origin is a minimum point

The origin is a minimum point if the Hessian matrix at the origin is positive definite. Since the Hessian matrix at the origin is a zero matrix, it is not positive definite, and hence the origin is not a minimum point.

This completes the analysis of the function $f(x, y, z) = x^4 + y^4 - z^6$.