Nabla

Definition

[math]\nabla(\ )=\mathbf{i}\frac{\partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\partial z}[/math]

Laplace operator

[math]\nabla\cdot\nabla(\ )=\nabla^2(\ )=\frac{\partial^2(\ )}{\partial x^2}+\frac{\partial^2(\ )}{\partial y^2}+\frac{\partial^2(\ )}{\partial z^2}[/math]

Notes (other forms seen)

I've seen a book (Vector Analysis and Cartesian Tensors - Third Edition - D E Borune & P C Kendall - which is a good book) distinguishbetween the [math]\nabla[/math]s used.

I will use [math]\vec\nabla[/math] to denote "bold" [math]\nabla[/math], which I usually draw by drawing a triangle, then a line down the left and across the top. I write just [math]\nabla[/math] as a triangle with a line down the left side. This works well.

I define [math]\vec\nabla^n(\ )=\mathbf{i}\frac{\partial^n(\ )}{\partial x^n}+\mathbf{j}\frac{\partial^n(\ )}{\partial y^n}+\mathbf{k}\frac{\partial^n(\ )}{\partial z^n}[/math] and [math]\nabla^n(\ )=\frac{\partial^n(\ )}{\partial x^n}+\frac{\partial^n(\ )}{\partial y^n}+\frac{\partial^n(\ )}{\partial z^n}[/math] as a slight extension to this notation.

1 book using this doesn't mean that the other books are wrong, it could be on to something. However in practice I have never actually come across the need for this. Which is why I list the first two definitions. I write this to show I have considered alternatives and why I do not use them.