Del operator and gradient


Vector differential operator `del’ is represented by a symbol \nabla. Its another name is `nabla’. For three dimensional case, it is defined as

\nabla = id/dx + jd/dy + kd/dz

where I, j, k are unit  vectors in the directions of co-ordinate axis X, Y and Z respectively.

Del is operated in three ways viz, gradient, divergence and curl, which we will discuss in next three sections.


The gradient of a  continuously differentiable scalar function v is defined as

grad v (\nabla v)= idv/dx + jdv/dy +kdv/dz

where i, j, k are the unit vectors along X, Y and Z-axes respectively,

Hence, the gradient of a scalar at any point in a scalar field is a vector.

Lamellar Vector Field Or Non-curl Field Or Conservative Field

A vector field derived from a scalar field by  taking its gradient is called lamellar  vector field or non-curl or conservative field. An electrostatic field, E is a lamellar field because it can be expressed as (negative)  gradient of scalar potential, V that is,

E = -grad V = – (idv/dx + jdv/dy +kdv/dz)

Note:After reading this article, you can answer the following questions:

  • Define del operator
  • What is the significance of del operator
  • name the three operations of del
  • What is gradient of scalar
  • What is the significance of gradient
  • Define lamellar vector field
  • Define non-curl field
  • Write the example of lamellar vector field.

I will discuss the del another two operations that is divergence and curl in the next articles.

Reference: This article is referred from my authored book “concepts of electromagnetic field theory” having ISBN 978-81-272-5245-8. In case of any doubt in this article or any other EMFT or physics related article, kindly post in the comment section.

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