Math 2.0

Real functions Definition

Scalar potential

Builds on Vector fieldGradientConservative vector field

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let F ⁣:U→ℝnF \colon U \to \mathbb{R}^n be a continuous vector field. A scalar potential of FF is a function ϕ ⁣:U→ℝ\phi \colon U \to \mathbb{R} of class C1C^1 with ∇ϕ=F\nabla \phi = F on UU.

Remarks

In physics, a potential is often a function VV with F=−∇VF = -\nabla V, so that VV is a potential energy. The two conventions differ by a sign. If UU is connected, two scalar potentials of FF differ by a constant. Indeed, their difference has zero gradient on UU. On a connected open set UU one finds a potential by fixing p∈Up \in U and setting ϕ(x)=∫γF⋅d𝐫\phi(x) = \int_\gamma F \cdot d\mathbf{r} for a path γ\gamma from pp to xx. This works when FF is conservative. Example: the gravitational field F(x)=−x/∥x∥3F(x) = -x / \|x\|^3 on ℝ3∖{0}\mathbb{R}^3 \setminus \{0\} has the scalar potential ϕ(x)=1/∥x∥\phi(x) = 1 / \|x\|.

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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…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptContinuity 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 derivativedefinitionGradient, definition, no evidence levelGradientdefinitionDifferentiable 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 fielddefinitionConservative vector field, definition, no evidence levelConservative vector f…definitionScalar potential, definition, no evidence levelScalar potentialdefinition
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  1. Euclidean space object
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  4. Function of several variables concept
  5. Vector-valued function concept
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  10. Partial derivative definition
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  12. Differentiable on an open set definition
  13. Higher partial derivatives definition
  14. Iterated derivative definition
  15. C^k function characterised by continuous derivatives statement
  16. Vector field definition
  17. Conservative vector field definition
  18. Scalar potential definition

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Scalar_potential, lead section
  2. Web https://en.wikipedia.org/wiki/Conservative_vector_field, Definition
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Tag0FP4BVDAAQ
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Kinddefinition
WritingW1 stated
MSC26B12
DefinesScalar potential
UsesVector field, Gradient, Conservative vector field

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