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Mathematics for Physics

214 Vector Differential Operator

A mapping that forms derivatives of scalar fields and of vector fields that is used to compute the gradient and related operators.

definition [d] (Vector Differential Operator = Del = Nabla = Gradient Operator) The symbolic vector operator \(\nabla\) that differentiates scalar and vector fields with respect to position. In Cartesian coordinates \((x, y, z)\):

  • \(\displaystyle \nabla \equiv \hat{\mathbf{x}}\, \dfrac{\partial}{\partial x} + \hat{\mathbf{y}}\, \dfrac{\partial}{\partial y} + \hat{\mathbf{z}}\, \dfrac{\partial}{\partial z}\) .

where

  • \(\nabla\) is the vector differential operator.
  • \(\hat{\mathbf{x}}, \hat{\mathbf{y}}, \hat{\mathbf{z}}\) are the Cartesian unit vectors.
  • \(x, y, z\) are Cartesian coordinates.
  • \(\partial / \partial x\), \(\partial / \partial y\), \(\partial / \partial z\) are partial derivative operators.

Note:

  • \(\nabla\) is not an ordinary vector, but an instruction to differentiate.
  • acting on a scalar \(f\) gives \(\nabla f\), the gradient; on a vector \(\mathbf{v}\) gives \(\nabla \cdot \mathbf{v}\), the divergence, and \(\nabla \times \mathbf{v}\), the curl.

214.1 Examples

example 1 [d] (Gradient — Riley et al.) The operator \(\nabla\) with

  • input \(\phi = xy^{2}z^{3}\) .

acts via the partial derivatives

  • \(\dfrac{\partial\phi}{\partial x} = y^{2}z^{3}\), \(\dfrac{\partial\phi}{\partial y} = 2xyz^{3}\), \(\dfrac{\partial\phi}{\partial z} = 3xy^{2}z^{2}\) ,

giving

  • \(\nabla\phi = y^{2}z^{3}\,\hat{\mathbf{i}} + 2xyz^{3}\,\hat{\mathbf{j}} + 3xy^{2}z^{2}\,\hat{\mathbf{k}}\) .

example 2 [d] (Divergence — Riley et al.) The operator \(\nabla\cdot\) with

  • input \(\mathbf{a} = x^{2}y^{2}\,\hat{\mathbf{i}} + y^{2}z^{2}\,\hat{\mathbf{j}} + x^{2}z^{2}\,\hat{\mathbf{k}}\) .

acts via the partial derivatives

  • \(\dfrac{\partial a_{x}}{\partial x} = 2xy^{2}\), \(\dfrac{\partial a_{y}}{\partial y} = 2yz^{2}\), \(\dfrac{\partial a_{z}}{\partial z} = 2x^{2}z\) ,

giving

  • \(\nabla\cdot\mathbf{a} = 2xy^{2} + 2yz^{2} + 2x^{2}z = 2(xy^{2} + yz^{2} + x^{2}z)\) .

example 3 [d] (Curl — Kreyszig) The operator \(\nabla\times\) with

  • input \(\mathbf{F} = [y,\, z,\, x]\) , so \(F_{x} = y\), \(F_{y} = z\), \(F_{z} = x\) .

acts via the partial derivatives

  • \(\dfrac{\partial F_{z}}{\partial y} - \dfrac{\partial F_{y}}{\partial z} = 0 - 1 = -1\) ,
  • \(\dfrac{\partial F_{x}}{\partial z} - \dfrac{\partial F_{z}}{\partial x} = 0 - 1 = -1\) ,
  • \(\dfrac{\partial F_{y}}{\partial x} - \dfrac{\partial F_{x}}{\partial y} = 0 - 1 = -1\) ,

giving

  • \(\nabla\times\mathbf{F} = [-1,\, -1,\, -1]\) .

214.2 References

  1. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — del / nabla; gradient, divergence, curl.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — vector differential operator.
  3. Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011. — nabla operator; curl example \(\mathbf{F}=[y,z,x]\mapsto[-1,-1,-1]\).
  4. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. Cambridge University Press, 2006. — del operator; gradient of \(\phi=xy^{2}z^{3}\); divergence of \(\mathbf{a}=x^{2}y^{2}\,\hat{\mathbf{i}}+y^{2}z^{2}\,\hat{\mathbf{j}}+x^{2}z^{2}\,\hat{\mathbf{k}}\).