Nabla

Definition

$$\nabla(\ )=\mathbf{i}\frac{\partial(\ )}{\partial x}+\mathbf{j}\frac{\partial(\ )}{\partial y}+\mathbf{k}\frac{\partial(\ )}{\partial z}$$

Laplace operator

$$\nabla\cdot\nabla(\ )=\nabla^2(\ )=\frac{\partial^2(\ )}{\partial x^2}+\frac{\partial^2(\ )}{\partial y^2}+\frac{\partial^2(\ )}{\partial z^2}$$

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 $$\nabla$$ s used.

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

I define $$\vec\nabla^n(\ )=\mathbf{i}\frac{\partial^n(\ )}{\partial x^n}+\mathbf{j}\frac{\partial^n(\ )}{\partial y^n}+\mathbf{k}\frac{\partial^n(\ )}{\partial z^n}$$ and $$\nabla^n(\ )=\frac{\partial^n(\ )}{\partial x^n}+\frac{\partial^n(\ )}{\partial y^n}+\frac{\partial^n(\ )}{\partial z^n}$$ 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.