Laplace equation: What is the difference between divergence operator and gradient operator both represented by $\nabla$? (in wikipedia article)

Question

Here is the beginning of the article of Laplace equation of wikipedia

Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as $$ \nabla^2f = 0\quad\text{or}\quad\Delta f = 0 $$ where $\Delta= \nabla \cdot \nabla = \nabla^2$ is the Laplace operator and $\nabla$ is divergence operator (also symbolized "div"), $\nabla$ is the gradient operator (also symbolized "grad"), and $f(x,y,z)$ is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function

My question: In this definition, what is the difference between divergence operator and gradient operator both represented by $\nabla$?

Answer 1

You left out a tiny detail. The wikipedia article states $\nabla\cdot$ is the divergence operator. Along with the dot product applied to a vector yields the divergence. $\nabla$ itself applied to some function is the classical gradient vector.

Answer 2

They are not both represented by $\nabla$: the gradient is $\nabla$ or sometimes $\vec{\nabla}$, the divergence is $\nabla\cdot$ or sometimes $\vec{\nabla}\cdot$

That little dot is a scalar product.

The whole thing is somewhat of an abuse of notation, which works if you imagine $\nabla$ as a vector of derivative operators, and then use the standard rules for the dot product. $$\nabla=[\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}]$$ $$\nabla f=[\frac{\partial}{\partial x} f,\frac{\partial}{\partial y} f,\frac{\partial}{\partial z} f]$$ $$\nabla \cdot [f_x,f_y,f_z]=\frac{\partial}{\partial x} f_x + \frac{\partial}{\partial y} f_y + \frac{\partial}{\partial z} f_z$$

Answer 3

For functions of three variables, $\nabla$ is shorthand for the following symbolic "vector": $$\nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)$$ You can use this with some flexibility, depending on how you let it act on a function:

• Gradient of a scalar function $f$ gives the following vector function: $$\nabla f = \left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right)$$
• Divergence of a vector function gives the following scalar function: $$\begin{align}\nabla \cdot \vec F &= \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right) \cdot \left(F_x,F_y,F_z\right)\\[5pt] & = \frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}\end{align}$$

Notice the scalar product in the notation for the divergence: $\nabla \color{blue}{\cdot} \vec F$.

And then you also have the Rotation or Curl of a vector function (giving another vector function) which can be expressed by the same operator, by using the cross product: $\nabla \times \vec F$.