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Mathematics for Physics

311 Derivation of Curl

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

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

\[ X = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \]

\[ Y = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \]

\[ Z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \]

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 transform together as Cartesian vector components. Therefore,

\[ \mathbf{R} = X\mathbf{i} + Y\mathbf{j} + Z\mathbf{k} \]

where

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

Substituting the differential components gives

\[ \mathbf{R} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k} \]

where

  • \(\mathbf{i}\) gives the \(x\) direction.
  • \(\mathbf{j}\) gives the \(y\) direction.
  • \(\mathbf{k}\) gives the \(z\) direction.

Curl. This vector is the curl of \(\mathbf{F}\):

\[ \operatorname{curl}\mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k} \]

where

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

Thus,

\[ (\operatorname{curl}\mathbf{F})_x = \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \]

\[ (\operatorname{curl}\mathbf{F})_y = \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \]

\[ (\operatorname{curl}\mathbf{F})_z = \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \]

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.

Cross 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 cross product is

\[ \mathbf{a}\times\mathbf{b} = (a_2b_3-a_3b_2)\mathbf{i} + (a_3b_1-a_1b_3)\mathbf{j} + (a_1b_2-a_2b_1)\mathbf{k} \]

where

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

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 cross-product rule gives

\[ \nabla\times\mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k} \]

where

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

Comparing with the component form,

\[ \operatorname{curl}\mathbf{F} = \nabla\times\mathbf{F}. \]

Determinant Form. The ordinary cross product may be written

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

where

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

Replacing \(\mathbf{a}\) by \(\nabla\) and \(\mathbf{b}\) by \(\mathbf{F}\) gives

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

where

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

Expanding the determinant,

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

Evaluating the minors gives

\[ \nabla\times\mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k}. \]

Rearranging the \(y\)-component,

\[ \nabla\times\mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\mathbf{k}. \]

Hence,

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

311.1 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.