Linked from

The 46 pages that link to Gradient, each with the reason it gives.

• Vector fieldRelated: It turns a scalar field into a vector field that points uphill.

• Gravitational potentialNarrower topic: Taking the gradient of potential determines the local gravitational field.

• Heat equationRelated: Temperature gradients drive conductive heat flux in Fourier's law.

• Poisson's equationRelated: Taking its divergence produces the Laplacian in Poisson's equation.

• Jacobian matrixCompared with: For scalar outputs, the Jacobian is the gradient written as a row under the usual convention.

• Partial derivativeNarrower topic: Its components collect all first-order rates of change at a point.

• ElectrostaticsRelated: The electric field is the negative spatial gradient of electric potential.

• Lagrange multiplierNarrower topic: At a constrained extremum, the objective gradient aligns with the constraint gradient.

• Tangent spaceRelated: A metric converts a function's differential into a tangent vector.

• Scalar fieldBroader topic: The gradient turns local changes in a scalar field into a direction and rate.

• DivergenceCompared with: Unlike divergence, it acts on a scalar field and returns a vector.

• VectorRelated: It assigns a direction and magnitude to local change in a scalar quantity.

• Gradient descentNarrower topic: Its negative direction determines each gradient descent update.

• Directional derivativeRelated: For differentiable functions, its dot product with a unit direction gives the directional derivative.

• Hessian matrixRelated: The Hessian is the Jacobian matrix of the gradient.

• Dual spaceCompared with: The derivative is naturally a covector; an inner product is needed to identify it with a gradient vector.

• Fick's laws of diffusionNarrower topic: Concentration gradients supply the spatial driving term in the first law.

• LaplacianRelated: The Laplacian begins by taking the gradient of the scalar field.

• Material derivativeRelated: Its directional variation combines with velocity to produce convective change.

• Pressure gradientNarrower topic: Pressure gradient is the gradient operator applied to the pressure field.

• Conservative forceRelated: Its relation to potential energy expresses the local direction and strength of the force.

• CurlCompared with: Unlike curl, the gradient acts on scalar fields and measures directional change.

• Pressure-gradient forceNarrower topic: The pressure gradient’s direction and steepness determine the force’s direction and magnitude.

• Vector calculusBroader topic: It converts scalar-field change into a direction and rate of increase.

• Karush–Kuhn–Tucker conditionsRelated: Objective and constraint gradients enter the stationarity relation.

• Neumann boundary conditionNarrower topic: The normal derivative is the gradient’s component perpendicular to the boundary.

• Equipotential surfaceRelated: The electric field is the negative gradient of electric potential.

• Exterior derivativeBroader topic: On functions, the exterior derivative produces the differential that corresponds to the gradient.

• Level setRelated: Where nonzero, it is perpendicular to the function's level sets.

• Surface normalRelated: For an implicitly defined surface, its gradient supplies a perpendicular direction.

• Total derivativeRelated: The gradient combines with a path’s velocity to produce the function’s total rate of change.

• Homogeneous polynomialRelated: Differentiating a homogeneous polynomial produces structured relations among its derivatives.

• Transport phenomenaRelated: Gradients in velocity, temperature, and concentration drive the principal transport processes.

• Vector (physics)Broader topic: It turns spatial variation in a scalar, such as temperature, into a direction and rate.

• Conservative vector fieldRelated: A field is conservative precisely when it equals a scalar function’s gradient.

• Real-valued functionRelated: It extends the derivative to real-valued functions of several real variables.

• Numerical optimizationRelated: It indicates local directions of steepest increase and supports many update rules.

• Multivariable calculusBroader topic: It combines directional rates of change into one vector.

• Euclidean vectorRelated: The Euclidean inner product identifies a function's differential with a gradient vector.

• Helmholtz decompositionRelated: The irrotational component can be represented as the gradient of a scalar potential.

• Gradient theoremBroader topic: Integrating this field along a path tracks the scalar field’s change.

• EquipotentialRelated: The gradient of the potential is perpendicular to its equipotentials.

• Field lineRelated: Potential gradients determine the direction of many physical fields and thus their line patterns.

• Function of several real variablesRelated: The gradient summarizes the function's first-order change at a point.

• Interior extremum theoremRelated: Its vanishing expresses the necessary condition for an extremum in several variables.

• Jacobian matrix and determinantCompared with: The gradient handles one scalar output; the Jacobian also covers vector-valued outputs.