Return to Question

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Definition of rotation invariant

My book doesn't provide a precise definition of rotation invariant but I think this is defined as follows:

Let $R[F]=\mathbb{C^{\infty}(\mathbb{R}^2)}[F_{x}, F_{y}, F_{xx}, F_{xy}, F_{yy}, ...]$ be a polynomial ring of infinite variables and $R(F)$ be its field of fractions. Elements in $R(F)$ can be seen as functionals. For a element $H(F)\in R(F)$, we define that $H(F)$ is rotation invariant iff for any rotation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$around the origin and for any $f\in C^{\infty}(\mathbb{R}^2)$, $H(f\circ T)=H(f)\circ T$ holds.

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Definition of rotation invariant

My book doesn't provide a precise definition of rotation invariant but I think this is defined as follows:

Let $R[F]={C^{\infty}(\mathbb{R}^2)}[F_{x}, F_{y}, F_{xx}, F_{xy}, F_{yy}, ...]$ be a polynomial ring of infinite variables and $R(F)$ be its field of fractions. Elements in $R(F)$ can be seen as functionals. For a element $H(F)\in R(F)$, we define that $H(F)$ is rotation invariant iff for any rotation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$around the origin and for any $f\in C^{\infty}(\mathbb{R}^2)$, $H(f\circ T)=H(f)\circ T$ holds.

Definition of combinations

Let $S(F)$ be a field of fractions of the ring $C^{\infty}(\mathbb{R}^2)[\Delta F, ||\nabla F||, \det{H},(\nabla F, H \nabla F)]$. A combination of the four quantities means an element of $S(F)$.

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Definition of rotation invariant

My book doesn't provide a precise definition of rotation invariant but I think this is defined as follows:

Let $R[F]=\mathbb{R}[F_{x}, F_{y}, F_{xx}, F_{xy}, F_{yy}, ...]$ be a polynomial ring of infinite variables. Elements in $R[F]$ can be seen as functionals. For a element $H(F)\in R[F]$, we define that $H(F)$ is rotation invariant iff for any rotation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$around the origin and for any $f\in C^{\infty}(\mathbb{R}^2)$, $H(f\circ T)=H(f)\circ T$ holds.

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Definition of rotation invariant

My book doesn't provide a precise definition of rotation invariant but I think this is defined as follows:

Let $R[F]=\mathbb{C^{\infty}(\mathbb{R}^2)}[F_{x}, F_{y}, F_{xx}, F_{xy}, F_{yy}, ...]$ be a polynomial ring of infinite variables and $R(F)$ be its field of fractions. Elements in $R(F)$ can be seen as functionals. For a element $H(F)\in R(F)$, we define that $H(F)$ is rotation invariant iff for any rotation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$around the origin and for any $f\in C^{\infty}(\mathbb{R}^2)$, $H(f\circ T)=H(f)\circ T$ holds.

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Let $(x, y, f(x,y))$ be a surface in $\mathbb{R^3}$. It is written in a book without proof that all rotation invariant (rotating around $z$-axis) of $f$ are combinations of the following four quantities:

• $\Delta f = f_{xx}+f_{yy}$
• $||\nabla f||^2=f_x^2 + f_y ^2$
• $\det{H} = f_{xx} f_{yy} - f_{xy}^2$
• $(\nabla f, H \nabla f) = f_{xx}f_x^2 + 2f_{xy}f_xf_y+f_{yy}f_y^2$

How do I prove this?

Definition of rotation invariant

My book doesn't provide a precise definition of rotation invariant but I think this is defined as follows:

Let $R[F]=\mathbb{R}[F_{x}, F_{y}, F_{xx}, F_{xy}, F_{yy}, ...]$ be a polynomial ring of infinite variables. Elements in $R[F]$ can be seen as functionals. For a element $H(F)\in R[F]$, we define that $H(F)$ is rotation invariant iff for any rotation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$around the origin and for any $f\in C^{\infty}(\mathbb{R}^2)$, $H(f\circ T)=H(f)\circ T$ holds.