Math 2.0

Real functions Concept

Del operator

Builds on GradientDivergenceCurlLaplacian

Summary

The del operator ∇=(∂∂x1,…,∂∂xn)\nabla = \left( \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right) is a vector of partial derivative operators on ℝn\mathbb{R}^n. It gives a short notation for the gradient ∇f\nabla f, the divergence ∇⋅F\nabla \cdot F, the curl ∇×F\nabla \times F in ℝ3\mathbb{R}^3, and the Laplacian ∇2f=∇⋅∇f\nabla^2 f = \nabla \cdot \nabla f.

Remarks

The notation treats ∇\nabla like a vector, but it is an operator. For example, ∇⋅F\nabla \cdot F and F⋅∇F \cdot \nabla are different: the first is a function, and the second is the operator ∑iFi ∂/∂xi\sum_i F_i \, \partial / \partial x_i. The directional derivative of ff along vv is (v⋅∇)f=v⋅∇f(v \cdot \nabla) f = v \cdot \nabla f. The symbol ∇\nabla is called nabla.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptCurl, concept, no evidence levelCurlconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptDel operator, concept, no evidence levelDel operatorconcept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Total derivative concept
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Differentiable function of several variables concept
  7. Partial derivative concept
  8. Vector field concept
  9. Gradient concept
  10. Curl concept
  11. Divergence concept
  12. Laplacian concept
  13. Del operator concept

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1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Del, Definition
  2. Web https://en.wikipedia.org/wiki/Del, Notational uses
About this page
Tag0H6DP8CHND
URIm2://0H6DP8CHND
Aliasesnabla, del
Kindconcept
WritingW1 stated
MSC26B12
UsesGradient, Divergence, Curl, Laplacian
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