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Mathematics for Physics

320 Gradient

A gradient is a vector of derivatives of a function of several variables that is used to calculate the change in a scalar value within a neighborhood.

Note: Also written \(\operatorname{grad} f\). Also written \(\nabla f\). Also read nabla \(f\).

definition [d] (Gradient = grad = \(\nabla f\)) From Kreyszig: given a scalar function \(f\) that is defined and differentiable in a domain in \(3\)-space with Cartesian coordinates \(x\), \(y\), \(z\), the gradient of \(f\), denoted \(\operatorname{grad} f\) or \(\nabla f\), is the vector function

  • \(\displaystyle \operatorname{grad} f = \nabla f = \dfrac{\partial f}{\partial x}\, \mathbf{i} + \dfrac{\partial f}{\partial y}\, \mathbf{j} + \dfrac{\partial f}{\partial z}\, \mathbf{k}\) .

In bracket component form,

  • \(\displaystyle \operatorname{grad} f = \left[ \dfrac{\partial f}{\partial x},\, \dfrac{\partial f}{\partial y},\, \dfrac{\partial f}{\partial z} \right]\) .

where

  • \(f\) is a differentiable scalar function.
  • \(x\), \(y\), \(z\) are Cartesian coordinates.
  • \(\operatorname{grad} f\) and \(\nabla f\) denote the gradient of \(f\).
  • \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are the Cartesian unit vectors.