Just when I thought I escaped from that cotangent formula after DDG...
hlin1
Why is the symbol for the Hessian sometimes used for Laplacian? It seems rather confusing.
keenan
@ceviri There is no escape! The cotan formula is everywhere. :-)
keenan
@hlin Some people like to write $\nabla^2$ for the Laplacian, since the Laplacian can be expressed as the divergence of the gradient. The divergence and gradient can in turn each be written using the "nabla" symbol. So, $\Delta = \nabla \cdot \nabla$, which is reminiscent of $\nabla^2$. But then others like to use (\nabla^2) for the Hessian. To make matters even more confusing, some like to write (\Delta) to just mean "change," e.g., (\Delta u) to indicate a finite change in a quantity (u), rather than the Laplacian of a function (u). For this reason, it's very important when you write to clearly state up-front what all your conventions are. For instance, at the beginning of a document you might write something like:
"Note that we will use $\Delta$ to denote the Laplacian, and $\nabla^2$ to denote the Hessian throughout."
Something else that can be helpful with differential operators is to just write out the abbreviated name. It's not uncommon for people to just write "div $X$", "grad $u$", or "Hess $u$". Though oddly enough, I've never seen anyone write something like "Lap $u$"!
Just when I thought I escaped from that cotangent formula after DDG...
Why is the symbol for the Hessian sometimes used for Laplacian? It seems rather confusing.
@ceviri There is no escape! The cotan formula is everywhere. :-)
@hlin Some people like to write $\nabla^2$ for the Laplacian, since the Laplacian can be expressed as the divergence of the gradient. The divergence and gradient can in turn each be written using the "nabla" symbol. So, $\Delta = \nabla \cdot \nabla$, which is reminiscent of $\nabla^2$. But then others like to use (\nabla^2) for the Hessian. To make matters even more confusing, some like to write (\Delta) to just mean "change," e.g., (\Delta u) to indicate a finite change in a quantity (u), rather than the Laplacian of a function (u). For this reason, it's very important when you write to clearly state up-front what all your conventions are. For instance, at the beginning of a document you might write something like:
"Note that we will use $\Delta$ to denote the Laplacian, and $\nabla^2$ to denote the Hessian throughout."
Something else that can be helpful with differential operators is to just write out the abbreviated name. It's not uncommon for people to just write "div $X$", "grad $u$", or "Hess $u$". Though oddly enough, I've never seen anyone write something like "Lap $u$"!