# Blog of Veikko M.O.T. Nyfors, Hybrid Quantum ICT consultant

Quantum Mechanics demystified, a try

Project maintained by veikkonyfors
Hosted on GitHub Pages — Theme by mattgraham

## Nabla Operator

Nabla operator, $\nabla$, describes the gradient of a scalar field in multiple dimensions.

It is a mathematical vector operation, widely used in physics, e.g. in connection of Maxwell equations.

In three dimensions, as used in Maxwell equations, operation is as below:

$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$

Or

∇f(x,y,z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
= (f(x+1,y,z)-f(x,y,z), f(x,y+1,z)-f(x,y,z), f(x,y,z+1)-f(x,y,z))

So, if you would have a scalar field like below (in 2 dimensions only, 3 would be harder to present):

40 |
60 |
60 |
60 |
60 |

60 |
85 |
85 |
85 |
60 |

60 |
85 |
100 |
85 |
60 |

60 |
65 |
85 |
85 |
60 |

60 |
60 |
60 |
60 |
60 |

you would end up having a vector field as follows

(20,20) |
(0,25) |
(0,25) |
(0,25) |
(0,0) |

(25,0) |
(0,0) |
(0,15) |
(-25,0) |
(0,0) |

(25,22.5) |
(15,20.0) |
(-15,0.0) |
(-25,-20.0) |
(0,-22.5) |

(5,0) |
(-20,-5) |
(0,-25) |
(-25,-25) |
(0,0) |

(-10.0,0) |
(-22.5,0) |
(-22.5,0) |
(-22.5,0) |
(-10.0,0) |

As an image

Shortly said, nabla tells into which direction scalar field is changing at each monitored spot.