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Mathematics for Physics

313 Derivation of Grad

\(\displaystyle \operatorname{grad}\mathbf{F} = J_{\mathbf{F}} = \begin{pmatrix} \dfrac{\partial F_{x}}{\partial x} & \dfrac{\partial F_{x}}{\partial y} & \dfrac{\partial F_{x}}{\partial z} \\ \dfrac{\partial F_{y}}{\partial x} & \dfrac{\partial F_{y}}{\partial y} & \dfrac{\partial F_{y}}{\partial z} \\ \dfrac{\partial F_{z}}{\partial x} & \dfrac{\partial F_{z}}{\partial y} & \dfrac{\partial F_{z}}{\partial z} \end{pmatrix}\)

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

\[ (\operatorname{grad}\mathbf{F})_{11} = \frac{\partial F_x}{\partial x} \]

\[ (\operatorname{grad}\mathbf{F})_{12} = \frac{\partial F_x}{\partial y} \]

\[ (\operatorname{grad}\mathbf{F})_{13} = \frac{\partial F_x}{\partial z} \]

\[ (\operatorname{grad}\mathbf{F})_{21} = \frac{\partial F_y}{\partial x} \]

\[ (\operatorname{grad}\mathbf{F})_{22} = \frac{\partial F_y}{\partial y} \]

\[ (\operatorname{grad}\mathbf{F})_{23} = \frac{\partial F_y}{\partial z} \]

\[ (\operatorname{grad}\mathbf{F})_{31} = \frac{\partial F_z}{\partial x} \]

\[ (\operatorname{grad}\mathbf{F})_{32} = \frac{\partial F_z}{\partial y} \]

\[ (\operatorname{grad}\mathbf{F})_{33} = \frac{\partial F_z}{\partial z} \]

where

  • \((\operatorname{grad}\mathbf{F})_{ij}\) is the differential component in row \(i\) and column \(j\).
  • \(x\), \(y\), \(z\) are Cartesian coordinates.
  • \(\partial\) denotes partial differentiation.

These quantities form a Cartesian matrix of components. Therefore,

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

where

  • \(\mathbf{R}\) is the resulting matrix field.
  • Each entry is a Cartesian differential component.

Substituting the differential components gives

\[ \mathbf{R} = \begin{pmatrix} (\operatorname{grad}\mathbf{F})_{11} & (\operatorname{grad}\mathbf{F})_{12} & (\operatorname{grad}\mathbf{F})_{13} \\ (\operatorname{grad}\mathbf{F})_{21} & (\operatorname{grad}\mathbf{F})_{22} & (\operatorname{grad}\mathbf{F})_{23} \\ (\operatorname{grad}\mathbf{F})_{31} & (\operatorname{grad}\mathbf{F})_{32} & (\operatorname{grad}\mathbf{F})_{33} \end{pmatrix}. \]

Position. Let

\[ \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} \]

and let a small change in position be

\[ \Delta\mathbf{x} = \begin{pmatrix} \Delta x \\ \Delta y \\ \Delta z \end{pmatrix}. \]

The corresponding change in the vector field is

\[ \Delta\mathbf{F} = \mathbf{F}(\mathbf{x}+\Delta\mathbf{x}) - \mathbf{F}(\mathbf{x}). \]

Writing each component,

\[ \Delta\mathbf{F} = \begin{pmatrix} F_x(x+\Delta x,y+\Delta y,z+\Delta z)-F_x(x,y,z) \\ F_y(x+\Delta x,y+\Delta y,z+\Delta z)-F_y(x,y,z) \\ F_z(x+\Delta x,y+\Delta y,z+\Delta z)-F_z(x,y,z) \end{pmatrix}. \]

First Component. For the \(x\)-component,

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

For small changes, the first-order change is

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

Second Component. Similarly,

\[ \Delta F_y = F_y(x+\Delta x,y+\Delta y,z+\Delta z) - F_y(x,y,z) \]

and

\[ \Delta F_y \approx \frac{\partial F_y}{\partial x}\Delta x + \frac{\partial F_y}{\partial y}\Delta y + \frac{\partial F_y}{\partial z}\Delta z. \]

Third Component. Similarly,

\[ \Delta F_z = F_z(x+\Delta x,y+\Delta y,z+\Delta z) - F_z(x,y,z) \]

and

\[ \Delta F_z \approx \frac{\partial F_z}{\partial x}\Delta x + \frac{\partial F_z}{\partial y}\Delta y + \frac{\partial F_z}{\partial z}\Delta z. \]

Vector Change. Combining the three component changes gives

\[ \Delta\mathbf{F} \approx \begin{pmatrix} \dfrac{\partial F_x}{\partial x}\Delta x + \dfrac{\partial F_x}{\partial y}\Delta y + \dfrac{\partial F_x}{\partial z}\Delta z \\[6pt] \dfrac{\partial F_y}{\partial x}\Delta x + \dfrac{\partial F_y}{\partial y}\Delta y + \dfrac{\partial F_y}{\partial z}\Delta z \\[6pt] \dfrac{\partial F_z}{\partial x}\Delta x + \dfrac{\partial F_z}{\partial y}\Delta y + \dfrac{\partial F_z}{\partial z}\Delta z \end{pmatrix}. \]

Gradient. This matrix is the gradient of \(\mathbf{F}\):

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

where

  • \(\operatorname{grad}\mathbf{F}\) is the gradient of \(\mathbf{F}\).
  • Each entry is a Cartesian gradient component.

Thus,

\[ (\operatorname{grad}\mathbf{F})_{ij} = \frac{\partial F_i}{\partial x_j} \]

where

  • \((x_1,x_2,x_3)=(x,y,z)\).
  • \((F_1,F_2,F_3)=(F_x,F_y,F_z)\).

Matrix Multiplication. For

\[ \mathbf{a} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

and

\[ \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \]

where

  • \(\mathbf{a}\) is the first factor.
  • \(\mathbf{b}\) is the second factor.
  • \(a_{ij}\) are entries of \(\mathbf{a}\).
  • \(b_1\), \(b_2\), \(b_3\) are components of \(\mathbf{b}\).

The matrix product is

\[ \mathbf{a}\mathbf{b} = \begin{pmatrix} a_{11}b_1+a_{12}b_2+a_{13}b_3 \\ a_{21}b_1+a_{22}b_2+a_{23}b_3 \\ a_{31}b_1+a_{32}b_2+a_{33}b_3 \end{pmatrix} \]

where

  • \(\mathbf{a}\mathbf{b}\) is the matrix product.
  • Each row 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

\[ \mathbf{a}=\operatorname{grad}\mathbf{F} \]

and

\[ \mathbf{b}=\Delta\mathbf{x}. \]

The matrix-product rule gives

\[ \Delta\mathbf{F} \approx \operatorname{grad}\mathbf{F}\, \Delta\mathbf{x} \]

where

  • \(\operatorname{grad}\mathbf{F}\) supplies the differential components.
  • \(\Delta\mathbf{x}\) supplies the coordinate increments.

Comparing with the component form,

\[ \operatorname{grad}\mathbf{F} = \mathbf{R}. \]

Apply \(\nabla\) separately to each scalar component of \(\mathbf{F}\):

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

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

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

These three component gradients form the rows of the gradient matrix:

\[ \operatorname{grad}\mathbf{F} = \begin{pmatrix} (\nabla F_x)^{T} \\ (\nabla F_y)^{T} \\ (\nabla F_z)^{T} \end{pmatrix}. \]

Directional Change. Let

\[ \mathbf{u} = \begin{pmatrix} u_x \\ u_y \\ u_z \end{pmatrix} \]

be a unit vector.

A small displacement in this direction is

\[ \Delta\mathbf{x} = \Delta s\,\mathbf{u}. \]

Substituting into

\[ \Delta\mathbf{F} \approx \operatorname{grad}\mathbf{F}\, \Delta\mathbf{x} \]

gives

\[ \Delta\mathbf{F} \approx \operatorname{grad}\mathbf{F}\, \mathbf{u}\,\Delta s. \]

Dividing by \(\Delta s\),

\[ \frac{\Delta\mathbf{F}}{\Delta s} \approx \operatorname{grad}\mathbf{F}\,\mathbf{u}. \]

Taking the limit as \(\Delta s\to0\) gives

\[ D_{\mathbf{u}}\mathbf{F} = \operatorname{grad}\mathbf{F}\,\mathbf{u} \]

where

  • \(D_{\mathbf{u}}\mathbf{F}\) is the directional derivative of the vector field.
  • \(\mathbf{u}\) specifies the direction.

Jacobian Form. The ordinary derivative of a map may be written

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

where

  • \(\mathbf{F}:\mathbb{R}^{3}\to\mathbb{R}^{3}\) is a vector-valued map.
  • \(J_{\mathbf{F}}\) is the Jacobian matrix.

Replacing the Jacobian by the gradient matrix gives

\[ \operatorname{grad}\mathbf{F} = J_{\mathbf{F}}. \]

Hence,

\(\displaystyle \operatorname{grad}\mathbf{F} = J_{\mathbf{F}} = \begin{pmatrix} \dfrac{\partial F_{x}}{\partial x} & \dfrac{\partial F_{x}}{\partial y} & \dfrac{\partial F_{x}}{\partial z} \\ \dfrac{\partial F_{y}}{\partial x} & \dfrac{\partial F_{y}}{\partial y} & \dfrac{\partial F_{y}}{\partial z} \\ \dfrac{\partial F_{z}}{\partial x} & \dfrac{\partial F_{z}}{\partial y} & \dfrac{\partial F_{z}}{\partial z} \end{pmatrix}\)

313.1 References

  1. Anand, L., & Govindjee, S. Continuum Mechanics of Solids. Oxford University Press, 2020. Chapter 2, §§2.1–2.2, pp. 23–25 — directional derivatives of vector fields; the gradient of a vector field as a second-order tensor; \((\operatorname{grad}\mathbf{v})_{ij}=\partial v_i/\partial x_j\).

  2. Bower, A. F. Applied Mechanics of Solids. CRC Press, 2009. Appendix A.2.8 — gradient of a vector field as a tensor field and its Cartesian matrix with entries \(\partial v_i/\partial x_j\).

  3. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 5th ed. Matrix Editions, 2015. §1.7 — derivative of a map between finite-dimensional spaces and the Jacobian matrix formed from its first partial derivatives.

  4. Kreyszig, E. Advanced Engineering Mathematics. 10th ed. Wiley, 2011. §9.4, p. 375 onward — vector and scalar functions and partial derivatives of vector functions; §9.6, p. 392 onward — functions of several variables, total differentials, and chain rules.

  5. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. 3rd ed. Cambridge University Press, 2006. Chapter 10, especially §§10.1, 10.4, 10.6–10.7 — differentiation of vectors, vector functions of several variables, scalar and vector fields, and vector differential operators.