Calculus How To

Del Operator (Nabla operator)

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The Del Operator (also called the Nabla operator or the vector differential operator) is a mathematical operator (actually a collection of partial derivative operators) commonly used in vector calculus to find higher dimensional derivatives. An “operator” is similar to a function (Green, 1998).

Although it’s defined as a vector, it doesn’t have magnitude and so isn’t a “true” vector.
del operator

By itself, it doesn’t actually have any meaning at all (much like a lonely multiplication or addition symbol). It only has a meaning when combined with a scalar function or vector function.

Del Operator Uses

The del operator is used in various vector calculus operations. Depending on what expression the operator appears in, it may denote gradient of a scalar field, divergence of a vector field, curl of a vector field, or the Laplacian operator:

  • Gradient: ∇ f
  • Divergence (the operator equivalent of a scalar product of two vectors): ∇ x E
  • Curl (the vector product equivalent of divergence): ∇ · E
  • Laplacian: ∇ · ∇f = ∇2f

The del operator can be applied to a scalar function or a vector function. ∇ · F is a scalar function, and ∇ x F is a vector function.


Green, K. (1998). The Del Operator. Retrieved January 8, 2020 from:
Hyperphysics. Vector Calculus.
Ida, N. (2007). Engineering Electromagnetics. Springer.

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