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Mathematics for Physics

313 Derivation of Divergence

\(\displaystyle \operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F} = \dfrac{\partial F_{x}}{\partial x} + \dfrac{\partial F_{y}}{\partial y} + \dfrac{\partial F_{z}}{\partial z}\)

\[ \begin{align*} \mathbf{F} &= F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k} \end{align*} \]

Start from a Cartesian vector field

where

  • \(\mathbf{F}\) is a vector field.
  • \(F_x\) is the \(x\)-component of \(\mathbf{F}\).
  • \(F_y\) is the \(y\)-component of \(\mathbf{F}\).
  • \(F_z\) is the \(z\)-component of \(\mathbf{F}\).
  • \(\mathbf{i}\) is the unit vector along \(x\).
  • \(\mathbf{j}\) is the unit vector along \(y\).
  • \(\mathbf{k}\) is the unit vector along \(z\).

\[ \begin{align*} X &= \dfrac{\partial F_x}{\partial x} \\ Y &= \dfrac{\partial F_y}{\partial y} \\ Z &= \dfrac{\partial F_z}{\partial z} \end{align*} \]

Define the three Cartesian differential contributions

where

  • \(X\) is the differential component along \(x\).
  • \(Y\) is the differential component along \(y\).
  • \(Z\) is the differential component along \(z\).
  • \(x\), \(y\), \(z\) are Cartesian coordinates.
  • \(\partial\) denotes partial differentiation.

\[ \begin{align*} R &= X+Y+Z \end{align*} \]

Add the contributions into one scalar

where

  • \(R\) is the resulting scalar field.
  • \(X\), \(Y\), \(Z\) are its Cartesian contributions.

\[ \begin{align*} R &= \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Substitute the differential contributions

\[ \begin{align*} \text{box sides} &= \Delta x,\ \Delta y,\ \Delta z \end{align*} \]

Take the side lengths of a small rectangular box

\[ \begin{align*} \Delta V &= \Delta x\,\Delta y\,\Delta z \end{align*} \]

Form the volume of the box

\[ \begin{align*} \Delta A_x &= \Delta y\,\Delta z \end{align*} \]

Write the area of a face perpendicular to x

\[ \begin{align*} \Phi_x^{+} &= F_x(x+\Delta x,y,z)\,\Delta y\,\Delta z \end{align*} \]

Estimate the outward flux through the face at x plus Dx

\[ \begin{align*} \Phi_x^{-} &= - F_x(x,y,z)\,\Delta y\,\Delta z \end{align*} \]

Reverse the sign on the opposite face, whose outward normal points backward

\[ \begin{align*} \Delta\Phi_x &= \bigl[F_x(x+\Delta x,y,z)-F_x(x,y,z)\bigr] \Delta y\,\Delta z \end{align*} \]

Subtract to obtain the net outward flux through the two x-faces

\[ \begin{align*} \dfrac{\partial F_x}{\partial x} &= \lim_{\Delta x\to0} \dfrac{F_x(x+\Delta x,y,z)-F_x(x,y,z)}{\Delta x} \end{align*} \]

Recall the definition of the partial derivative

\[ \begin{align*} F_x(x+\Delta x,y,z)-F_x(x,y,z) &\approx \dfrac{\partial F_x}{\partial x}\Delta x \end{align*} \]

Replace the difference by its first-order increment

\[ \begin{align*} \Delta\Phi_x &\approx \dfrac{\partial F_x}{\partial x}\Delta V \\ \Delta\Phi_y &\approx \dfrac{\partial F_y}{\partial y}\Delta V \\ \Delta\Phi_z &\approx \dfrac{\partial F_z}{\partial z}\Delta V \end{align*} \]

Convert each pair of opposite faces into a volume times a partial derivative

\[ \begin{align*} \Delta\Phi &= \Delta\Phi_x+\Delta\Phi_y+\Delta\Phi_z \end{align*} \]

Add the three pairs to obtain the total outward flux

\[ \begin{align*} \Delta\Phi &\approx \biggl( \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \biggr) \Delta V \end{align*} \]

Substitute the three face contributions

\[ \begin{align*} \dfrac{\Delta\Phi}{\Delta V} &\approx \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Divide by the volume to obtain a flux density

\[ \begin{align*} \lim_{\Delta V\to0} \dfrac{\Delta\Phi}{\Delta V} &= \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Shrink the box to a point

\[ \begin{align*} \operatorname{div}\mathbf{F} &= \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Name this flux density the divergence of F

where

  • \(\operatorname{div}\mathbf{F}\) is the divergence of \(\mathbf{F}\).
  • Each term is a Cartesian differential contribution.

\[ \begin{align*} \operatorname{div}\mathbf{F} &= X+Y+Z \end{align*} \]

Compare with the assembled scalar

\[ \begin{align*} \mathbf{a} &= a_1\mathbf{i}+a_2\mathbf{j}+a_3\mathbf{k} \\ \mathbf{b} &= b_1\mathbf{i}+b_2\mathbf{j}+b_3\mathbf{k} \end{align*} \]

Write two general Cartesian vectors

where

  • \(\mathbf{a}\) is the first vector.
  • \(\mathbf{b}\) is the second vector.
  • \(a_1\), \(a_2\), \(a_3\) are components of \(\mathbf{a}\).
  • \(b_1\), \(b_2\), \(b_3\) are components of \(\mathbf{b}\).

\[ \begin{align*} \mathbf{a}\cdot\mathbf{b} &= a_1b_1+a_2b_2+a_3b_3 \end{align*} \]

Form the ordinary dot product

where

  • \(\mathbf{a}\cdot\mathbf{b}\) is the dot product.
  • Each term is a Cartesian contribution.

\[ \begin{align*} \nabla &= \mathbf{i}\dfrac{\partial}{\partial x} \\ &\quad +\mathbf{j}\dfrac{\partial}{\partial y} \\ &\quad +\mathbf{k}\dfrac{\partial}{\partial z} \end{align*} \]

Define the vector differential operator

where

  • \(\nabla\) is the vector differential operator.
  • \(\dfrac{\partial}{\partial x}\) differentiates with respect to \(x\).
  • \(\dfrac{\partial}{\partial y}\) differentiates with respect to \(y\).
  • \(\dfrac{\partial}{\partial z}\) differentiates with respect to \(z\).

\[ \begin{align*} \mathbf{a} &= \nabla \\ \mathbf{b} &= \mathbf{F} \end{align*} \]

Identify the dot-product factors with nabla and the field

\[ \begin{align*} \nabla\cdot\mathbf{F} &= \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Apply the dot-product rule

where

  • \(\nabla\) supplies the differential operators.
  • \(\mathbf{F}\) supplies the field components.
  • \(\cdot\) applies the dot-product rule.

\[ \begin{align*} \operatorname{div}\mathbf{F} &= \nabla\cdot\mathbf{F} \end{align*} \]

Compare with the component form

\[ \begin{align*} \operatorname{grad}\mathbf{F} &= \begin{pmatrix} \dfrac{\partial F_x}{\partial x} & \dfrac{\partial F_x}{\partial y} & \dfrac{\partial F_x}{\partial z} \\ \dfrac{\partial F_y}{\partial x} & \dfrac{\partial F_y}{\partial y} & \dfrac{\partial F_y}{\partial z} \\ \dfrac{\partial F_z}{\partial x} & \dfrac{\partial F_z}{\partial y} & \dfrac{\partial F_z}{\partial z} \end{pmatrix} \end{align*} \]

Write the Cartesian gradient matrix of F

where

  • Each row contains partial derivatives of one field component.
  • Each column contains partial derivatives along one coordinate.

\[ \begin{align*} \operatorname{tr}(\operatorname{grad}\mathbf{F}) &= \dfrac{\partial F_x}{\partial x} \\ &\quad + \dfrac{\partial F_y}{\partial y} \\ &\quad + \dfrac{\partial F_z}{\partial z} \end{align*} \]

Sum the diagonal entries

where

  • \(\operatorname{tr}\) denotes the trace of a matrix.
  • The trace is the sum of the diagonal entries.

\[ \begin{align*} \operatorname{div}\mathbf{F} &= \operatorname{tr}(\operatorname{grad}\mathbf{F}) \end{align*} \]

Compare with the component form

Hence,

\(\displaystyle \operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F} = \dfrac{\partial F_{x}}{\partial x} + \dfrac{\partial F_{y}}{\partial y} + \dfrac{\partial F_{z}}{\partial z}\)

313.1 References

  1. Kreyszig, E. Advanced Engineering Mathematics. 10th ed. Wiley, 2011. §9.8 — divergence of a vector field and its Cartesian representation.

  2. Kreyszig, E. Advanced Engineering Mathematics. 10th ed. Wiley, 2011. §10.7 — Gauss’s divergence theorem and the relation between divergence and outward flux.

  3. Griffiths, D. J. Introduction to Electrodynamics. 4th ed. Pearson, 2013. Chapter 1 — divergence, flux, and the divergence theorem.

  4. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. 3rd ed. Cambridge University Press, 2006. Chapters 10–11 — vector differential operators and surface and volume integrals.

  5. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists: A Comprehensive Guide. 7th ed. Academic Press, 2012. Chapter 3 — vector analysis, differential vector operators, surface integrals, and integral theorems.