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Mathematics for Physics

314 Derivation of Grad

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

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

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 \(z\).

\[ \begin{align*} (\operatorname{grad}\mathbf{F})_{11} &= \dfrac{\partial F_x}{\partial x} \\ (\operatorname{grad}\mathbf{F})_{12} &= \dfrac{\partial F_x}{\partial y} \\ (\operatorname{grad}\mathbf{F})_{13} &= \dfrac{\partial F_x}{\partial z} \\ (\operatorname{grad}\mathbf{F})_{21} &= \dfrac{\partial F_y}{\partial x} \\ (\operatorname{grad}\mathbf{F})_{22} &= \dfrac{\partial F_y}{\partial y} \\ (\operatorname{grad}\mathbf{F})_{23} &= \dfrac{\partial F_y}{\partial z} \\ (\operatorname{grad}\mathbf{F})_{31} &= \dfrac{\partial F_z}{\partial x} \\ (\operatorname{grad}\mathbf{F})_{32} &= \dfrac{\partial F_z}{\partial y} \\ (\operatorname{grad}\mathbf{F})_{33} &= \dfrac{\partial F_z}{\partial z} \end{align*} \]

Define the nine Cartesian differential components

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.

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

Assemble those components into a matrix

where

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

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

Rewrite the matrix in index form

\[ \begin{align*} \mathbf{x} &= \begin{pmatrix} x \\ y \\ z \end{pmatrix} \\ \Delta\mathbf{x} &= \begin{pmatrix} \Delta x \\ \Delta y \\ \Delta z \end{pmatrix} \end{align*} \]

Take a position and a small change in position

\[ \begin{align*} \Delta\mathbf{F} &= \mathbf{F}(\mathbf{x}+\Delta\mathbf{x})-\mathbf{F}(\mathbf{x}) \end{align*} \]

Form the corresponding change in the field

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

Write the change in each Cartesian component

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

Keep the first-order increment of each component

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

Name this matrix the gradient of F

where

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

\[ \begin{align*} (\operatorname{grad}\mathbf{F})_{ij} &= \dfrac{\partial F_i}{\partial x_j} \end{align*} \]

Write the entries in index notation

where

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

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

Write a general matrix and a general column vector

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 \(\mathbf{b}\).

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

Form the ordinary matrix product

where

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

\[ \begin{align*} \mathbf{a} &= \operatorname{grad}\mathbf{F} \\ \mathbf{b} &= \Delta\mathbf{x} \end{align*} \]

Identify the factors with the gradient matrix and the displacement

\[ \begin{align*} \Delta\mathbf{F} &\approx \operatorname{grad}\mathbf{F}\,\Delta\mathbf{x} \end{align*} \]

Apply the matrix-product rule

where

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

\[ \begin{align*} \operatorname{grad}\mathbf{F} &= \mathbf{R} \end{align*} \]

Compare with the assembled matrix

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

Apply nabla separately to each scalar component

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

Stack those component gradients as the rows of the matrix

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

Take a unit direction

\[ \begin{align*} \Delta\mathbf{x} &= \Delta s\,\mathbf{u} \end{align*} \]

Write a small displacement along that direction

\[ \begin{align*} \Delta\mathbf{F} &\approx \operatorname{grad}\mathbf{F}\,\mathbf{u}\,\Delta s \end{align*} \]

Substitute the directed displacement

\[ \begin{align*} \dfrac{\Delta\mathbf{F}}{\Delta s} &\approx \operatorname{grad}\mathbf{F}\,\mathbf{u} \end{align*} \]

Divide by the step length

\[ \begin{align*} D_{\mathbf{u}}\mathbf{F} &= \operatorname{grad}\mathbf{F}\,\mathbf{u} \end{align*} \]

Pass to the directional derivative

where

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

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

Write the Jacobian matrix of the map F

where

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

\[ \begin{align*} \operatorname{grad}\mathbf{F} &= J_{\mathbf{F}} \end{align*} \]

Identify the gradient matrix with the Jacobian

Hence,

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

314.1 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.