Math 2.0

Classical thermodynamics, heat transfer Definition

Anisotropic heat conduction

Builds on Thermal conductionFourier's law of heat conductionHeat fluxSymmetric matrixPositive definite matrixGradient

Definition

Let a body occupy an open set Ω⊆ℝn\Omega \subseteq \R^n, with temperature TT differentiable in xx. Let K(x)K(x) be a symmetric positive definite n×nn \times n matrix for each x∈Ωx \in \Omega, the conductivity tensor. Anisotropic heat conduction is conduction whose heat flux is 𝐪=−K ∇T,\mathbf{q} = -K \, \nabla T, where KK is not a scalar multiple of the identity matrix. With density ρ\rho, specific heat cc and source density ff, the temperature obeys ρ c ∂tT=∇⋅(K∇T)+f.\rho \, c \, \partial_t T = \nabla \cdot \left(K \nabla T\right) + f.

Remarks

When K=kIK = k I, this is the isotropic Fourier's law.

Example: in ℝ2\R^2 let K=diag⁡(4,1)K = \operatorname{diag}(4, 1) and ∇T=(1,1)\nabla T = (1, 1). Then 𝐪=(−4,−1)\mathbf{q} = (-4, -1), which is not parallel to ∇T\nabla T. Still 𝐪⋅∇T=−5<0\mathbf{q} \cdot \nabla T = -5 < 0, so heat flows to the colder side.

Non-example: K=diag⁡(1,−1)K = \operatorname{diag}(1, -1) is symmetric but not positive definite. With ∇T=(0,1)\nabla T = (0, 1) it gives 𝐪=(0,1)\mathbf{q} = (0, 1), so heat would flow from cold to hot.

Positive definiteness is the condition 𝐪⋅∇T=−∇T⋅K∇T<0\mathbf{q} \cdot \nabla T = -\nabla T \cdot K \nabla T < 0 whenever ∇T≠0\nabla T \ne 0.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHeat flux, concept, no evidence levelHeat fluxconceptPositive definite matrix, concept, no evidence levelPositive definite mat…conceptSymmetric matrix, concept, no evidence levelSymmetric matrixconceptTemperature, concept, no evidence levelTemperatureconceptThermal conduction, concept, no evidence levelThermal conductionconceptThermal conductivity, concept, no evidence levelThermal conductivityconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptGradient, concept, no evidence levelGradientconceptFourier's law of heat conduction, statement, no evidence levelFourier's law of heat…statementAnisotropic heat conduction, definition, no evidence levelAnisotropic heat cond…definition
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. Heat flux concept
  4. Positive definite matrix concept
  5. Symmetric matrix concept
  6. Temperature concept
  7. Thermal conduction concept
  8. Thermal conductivity concept
  9. Total derivative concept
  10. Function of several variables concept
  11. Differentiable function of several variables concept
  12. Partial derivative concept
  13. Gradient concept
  14. Fourier's law of heat conduction statement
  15. Anisotropic heat conduction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Thermal_conduction
About this page
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WritingW0 stub
MSC80A19, 80A17
DefinesAnisotropic heat conduction
UsesThermal conduction, Fourier's law of heat conduction, Heat flux, Symmetric matrix, Positive definite matrix, Gradient
Changed3 November 2026

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