203 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.