Math 2.0

Real functions Definition

Vector field

Builds on Vector-valued functionContinuity in R^nC^k function characterised by continuous derivatives

Definition

Let S⊆ℝnS \subseteq \mathbb{R}^n. A vector field on SS is a vector-valued function F ⁣:S→ℝnF \colon S \to \mathbb{R}^n, F(x)=(F1(x),…,Fn(x))F(x) = (F_1(x), \dots, F_n(x)). The vector field is continuous, or of class CkC^k, if each component FiF_i is continuous, or of class C^k, on SS.

Remarks

One pictures FF by an arrow F(x)F(x) drawn with its tail at the point xx. For a differentiable function f ⁣:U→ℝf \colon U \to \mathbb{R} on an open set UU, the gradient ∇f\nabla f is a vector field on UU. Such fields are gradient fields. The field F(x,y)=(−y,x)F(x, y) = (-y, x) on ℝ2\mathbb{R}^2 describes a counterclockwise rotation about the origin. The field F(x)=xF(x) = x points away from the origin. In ℝ3\mathbb{R}^3 one often writes F=P 𝐢+Q 𝐣+R 𝐤F = P \, \mathbf{i} + Q \, \mathbf{j} + R \, \mathbf{k}, with PP, QQ, RR the components of FF.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementVector field, definition, no evidence levelVector fielddefinition
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  1. Euclidean space object
  2. Function concept
  3. Fréchet derivative definition
  4. Function of several variables concept
  5. Vector-valued function concept
  6. Limit of a function of several variables concept
  7. Continuity in R^n definition
  8. Differentiable function of several variables definition
  9. Partial derivative definition
  10. Differentiable on an open set definition
  11. Higher partial derivatives definition
  12. Iterated derivative definition
  13. C^k function characterised by continuous derivatives statement
  14. Vector field definition

Where this leads

57 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Vector_field, Vector fields on subsets of Euclidean space
  2. Web https://en.wikipedia.org/wiki/Vector_field, Examples
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Kinddefinition
WritingW1 stated
MSC26B12
DefinesVector field
UsesVector-valued function, Continuity in R^n, C^k function characterised by continuous derivatives

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