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Mathematics for Physics

312 Derivation of Divergence

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

Vector Field. Let

\[ \mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k} \]

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 \(z\).

Differential Components. Define

\[ X = \frac{\partial F_x}{\partial x} \]

\[ Y = \frac{\partial F_y}{\partial y} \]

\[ Z = \frac{\partial F_z}{\partial z} \]

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.

These quantities add as Cartesian scalar contributions. Therefore,

\[ R = X + Y + Z \]

where

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

Substituting the differential components gives

\[ R = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. \]

Flux Through a Small Box. Consider a rectangular box with side lengths

\[ \Delta x, \qquad \Delta y, \qquad \Delta z. \]

Its volume is

\[ \Delta V = \Delta x\,\Delta y\,\Delta z. \]

The two faces perpendicular to the \(x\)-axis each have area

\[ \Delta A_x = \Delta y\,\Delta z. \]

The outward flux through the face at \(x+\Delta x\) is approximately

\[ F_x(x+\Delta x,y,z) \Delta y\,\Delta z. \]

The outward normal of the opposite face points in the negative \(x\)-direction, so its outward flux is approximately

$$
F_x(x,y,z)
y,z.
$$

Therefore, the net outward flux through the two \(x\)-faces is

\[ \begin{aligned} \Delta\Phi_x &= F_x(x+\Delta x,y,z) \Delta y\,\Delta z - F_x(x,y,z) \Delta y\,\Delta z \\ &= \left[ F_x(x+\Delta x,y,z) - F_x(x,y,z) \right] \Delta y\,\Delta z. \end{aligned} \]

By the definition of the partial derivative,

\[ \frac{\partial F_x}{\partial x} = \lim_{\Delta x\to0} \frac{ F_x(x+\Delta x,y,z) - F_x(x,y,z) }{ \Delta x }. \]

For small \(\Delta x\),

\[ F_x(x+\Delta x,y,z) - F_x(x,y,z) \approx \frac{\partial F_x}{\partial x}\Delta x. \]

Substituting gives

\[ \Delta\Phi_x \approx \frac{\partial F_x}{\partial x} \Delta x\,\Delta y\,\Delta z. \]

Since

\[ \Delta V = \Delta x\,\Delta y\,\Delta z, \]

we obtain

\[ \Delta\Phi_x \approx \frac{\partial F_x}{\partial x} \Delta V. \]

Similarly,

\[ \Delta\Phi_y \approx \frac{\partial F_y}{\partial y} \Delta V \]

and

\[ \Delta\Phi_z \approx \frac{\partial F_z}{\partial z} \Delta V. \]

Total Flux. The total outward flux through the box is

\[ \Delta\Phi = \Delta\Phi_x + \Delta\Phi_y + \Delta\Phi_z. \]

Substituting the three contributions gives

\[ \Delta\Phi \approx \left( \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \right) \Delta V. \]

Dividing by the volume gives

\[ \frac{\Delta\Phi}{\Delta V} \approx \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. \]

Taking the limit as the box shrinks to a point gives

\[ \lim_{\Delta V\to0} \frac{\Delta\Phi}{\Delta V} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}. \]

Divergence. This scalar is the divergence of \(\mathbf{F}\):

\[ \operatorname{div}\mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

where

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

Thus,

\[ \operatorname{div}\mathbf{F} = X + Y + Z. \]

Dot Product. For

\[ \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \]

and

\[ \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \]

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 \(\mathbf{b}\).

The dot product is

\[ \mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]

where

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

Differential Operator. Define

\[ \nabla = \mathbf{i}\frac{\partial}{\partial x} + \mathbf{j}\frac{\partial}{\partial y} + \mathbf{k}\frac{\partial}{\partial z} \]

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 \(z\).

Set

\[ \mathbf{a}=\nabla \]

and

\[ \mathbf{b}=\mathbf{F}. \]

The dot-product rule gives

\[ \nabla\cdot\mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

where

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

Comparing with the component form,

\[ \operatorname{div}\mathbf{F} = \nabla\cdot\mathbf{F}. \]

Trace Form. The ordinary gradient matrix may be written

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

where

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

The trace of this matrix is

\[ \operatorname{tr} \left( \operatorname{grad}\mathbf{F} \right) = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \]

where

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

Comparing with the component form,

\[ \operatorname{div}\mathbf{F} = \operatorname{tr} \left( \operatorname{grad}\mathbf{F} \right). \]

Hence,

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

312.1 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.