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Mathematics for Physics

312 Derivation of Curl

\(\displaystyle \operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{vmatrix}\)

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

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 \(z\).

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

Define the three Cartesian differential components

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.

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

Assemble those components into one vector

where

  • \(\mathbf{R}\) is the resulting vector field.
  • \(X\), \(Y\), \(Z\) are its Cartesian components.

\[ \begin{align*} \mathbf{R} &= \biggl(\dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z}\biggr)\mathbf{i} \\ &\quad +\biggl(\dfrac{\partial F_x}{\partial z}-\dfrac{\partial F_z}{\partial x}\biggr)\mathbf{j} \\ &\quad +\biggl(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\biggr)\mathbf{k} \end{align*} \]

Substitute the differential components

\[ \begin{align*} \operatorname{curl}\mathbf{F} &= \biggl(\dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z}\biggr)\mathbf{i} \\ &\quad +\biggl(\dfrac{\partial F_x}{\partial z}-\dfrac{\partial F_z}{\partial x}\biggr)\mathbf{j} \\ &\quad +\biggl(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\biggr)\mathbf{k} \end{align*} \]

Name this vector the curl of F

where

  • \(\operatorname{curl}\mathbf{F}\) is the curl of \(\mathbf{F}\).
  • Each coefficient is a Cartesian curl component.

\[ \begin{align*} (\operatorname{curl}\mathbf{F})_x &= \dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z} \\ (\operatorname{curl}\mathbf{F})_y &= \dfrac{\partial F_x}{\partial z}-\dfrac{\partial F_z}{\partial x} \\ (\operatorname{curl}\mathbf{F})_z &= \dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y} \end{align*} \]

Read off the three Cartesian curl components

where

  • \((\operatorname{curl}\mathbf{F})_x\) is the \(x\)-component.
  • \((\operatorname{curl}\mathbf{F})_y\) is the \(y\)-component.
  • \((\operatorname{curl}\mathbf{F})_z\) is the \(z\)-component.

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

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 \(\mathbf{b}\).

\[ \begin{align*} \mathbf{a}\times\mathbf{b} &= (a_2b_3-a_3b_2)\mathbf{i} \\ &\quad +(a_3b_1-a_1b_3)\mathbf{j} \\ &\quad +(a_1b_2-a_2b_1)\mathbf{k} \end{align*} \]

Form the cross product

where

  • \(\mathbf{a}\times\mathbf{b}\) is the cross product.
  • Each coefficient is a Cartesian component.

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

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 cross-product factors with nabla and the field

\[ \begin{align*} \nabla\times\mathbf{F} &= \biggl(\dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z}\biggr)\mathbf{i} \\ &\quad +\biggl(\dfrac{\partial F_x}{\partial z}-\dfrac{\partial F_z}{\partial x}\biggr)\mathbf{j} \\ &\quad +\biggl(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\biggr)\mathbf{k} \end{align*} \]

Apply the cross-product rule

\[ \begin{align*} \operatorname{curl}\mathbf{F} &= \nabla\times\mathbf{F} \end{align*} \]

Compare with the component form

where

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

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

Write the cross product as a determinant

where

  • The first row contains Cartesian unit vectors.
  • The second row contains \(\mathbf{a}\) components.
  • The third row contains \(\mathbf{b}\) components.

\[ \begin{align*} \nabla\times\mathbf{F} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix} \end{align*} \]

Replace the two vectors by nabla and the field

where

  • The first row contains Cartesian unit vectors.
  • The second row contains differential operators.
  • The third row contains field components.
  • The determinant is symbolic.

\[ \begin{align*} \nabla\times\mathbf{F} &= \mathbf{i} \begin{vmatrix} \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_y & F_z \end{vmatrix} \\ &\quad - \mathbf{j} \begin{vmatrix} \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial z} \\ F_x & F_z \end{vmatrix} \\ &\quad + \mathbf{k} \begin{vmatrix} \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} \\ F_x & F_y \end{vmatrix} \end{align*} \]

Expand the determinant

\[ \begin{align*} \nabla\times\mathbf{F} &= \biggl(\dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z}\biggr)\mathbf{i} \\ &\quad - \biggl(\dfrac{\partial F_z}{\partial x}-\dfrac{\partial F_x}{\partial z}\biggr)\mathbf{j} \\ &\quad + \biggl(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\biggr)\mathbf{k} \end{align*} \]

Evaluate the minors

\[ \begin{align*} \nabla\times\mathbf{F} &= \biggl(\dfrac{\partial F_z}{\partial y}-\dfrac{\partial F_y}{\partial z}\biggr)\mathbf{i} \\ &\quad + \biggl(\dfrac{\partial F_x}{\partial z}-\dfrac{\partial F_z}{\partial x}\biggr)\mathbf{j} \\ &\quad + \biggl(\dfrac{\partial F_y}{\partial x}-\dfrac{\partial F_x}{\partial y}\biggr)\mathbf{k} \end{align*} \]

Rearrange the y-component

Hence,

\(\displaystyle \operatorname{curl} \mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{vmatrix}\)

A rectangular loop in the yz-plane with corners (0,y,z), (0,y+\Delta y,z), (0,y+\Delta y,z+\Delta z), and (0,y,z+\Delta z). The path is traversed counterclockwise as seen from the positive x-axis.
A rectangular loop in the \(yz\)-plane with corners \((0,y,z)\), \((0,y+\Delta y,z)\), \((0,y+\Delta y,z+\Delta z)\), and \((0,y,z+\Delta z)\). The path is traversed counterclockwise as seen from the positive \(x\)-axis.

312.1 References

  1. MacCullagh, J. An Essay towards a Dynamical Theory of Crystalline Reflexion and Refraction. Transactions of the Royal Irish Academy, Vol. XXI. Read 9 December 1839; published 1846.

  2. Lunney, J. G. James MacCullagh. Trinity College Dublin, 2015.

  3. Kreyszig, E. Advanced Engineering Mathematics, 10th ed. Wiley, 2011.

  4. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering.