Linked from

The 21 pages that link to Laplacian, each with the reason it gives.

• GradientCompared with: It applies another derivative after the gradient, measuring aggregate local curvature.

• Heat equationRelated: The spatial Laplacian measures the local curvature that drives temperature change.

• Poisson's equationBroader topic: It measures the local curvature on the left side of Poisson's equation.

• Laplace's equationBroader topic: Setting this operator to zero is the equation's defining condition.

• Scalar fieldRelated: It detects local curvature and appears in equations governing scalar fields.

• DivergenceCompared with: It combines spatial second derivatives rather than directly measuring a vector field’s flux.

• Hessian matrixCompared with: Unlike the full Hessian, it reduces curvature information to a scalar.

• Harmonic functionRelated: Its vanishing is the defining condition for a harmonic function.

• Maximum principleRelated: The classical maximum principle is most familiar for equations defined by this operator.

• CurlCompared with: The Laplacian measures scalar or componentwise spreading rather than circulation.

• Reaction–diffusion systemRelated: The Laplacian encodes how concentration differences drive spatial smoothing.

• Vector calculusRelated: It combines two central operators to describe local spatial curvature.

• Diffusion equationBroader topic: The Laplacian measures local curvature that drives diffusion in the standard equation.

• Helmholtz equationNarrower topic: The equation is built by adding a constant multiple of the unknown to this operator.

• Elliptic operatorRelated: The Laplacian is the prototype against which many elliptic equations are introduced.

• Nonlinear Schrödinger equationRelated: Its spatial action represents diffraction or dispersion in common forms of the equation.

• Helmholtz decompositionRelated: Taking divergence or curl of the decomposition leads to Poisson equations involving the Laplacian.

• Quantum potentialRelated: Applying it to the amplitude captures the curvature in the potential’s formula.

• Faxén's lawRelated: A Laplacian correction captures how velocity varies across the sphere.

• Harnack's principleRelated: A harmonic function is defined by the vanishing of this operator.

• Weyl's lemmaRelated: Weyl's lemma assumes this operator annihilates the distribution.