Vector Calculus Identities
A Complete Reference
This page collects the standard identities of vector calculus in three-dimensional Euclidean space $\mathbb{R}^3$. Throughout, $f, g$ denote scalar fields and $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$ denote vector fields.
1. Algebra of the Dot and Cross Products
1.1 Basic properties
(1) Commutativity of the dot product Proof
$$\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \tag{1}$$(2) Anticommutativity of the cross product Proof
$$\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a} \tag{2}$$(3) Distributivity of the cross product Proof
$$\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \tag{3}$$(4) Compatibility with scalar multiplication Proof
$$(c\,\mathbf{a}) \times \mathbf{b} = \mathbf{a} \times (c\,\mathbf{b}) = c\,(\mathbf{a} \times \mathbf{b}) \tag{4}$$1.2 Scalar triple product
(5) Cyclic symmetry of the scalar triple product Proof
$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c} \times \mathbf{a}) = \mathbf{c} \cdot (\mathbf{a} \times \mathbf{b}) \tag{5}$$(6) Antisymmetry of the scalar triple product Proof
$$\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = -\mathbf{a} \cdot (\mathbf{c} \times \mathbf{b}) \tag{6}$$1.3 Vector triple product (BAC-CAB rule)
(7) The BAC-CAB rule Proof
$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) \tag{7}$$(8) Right-grouped vector triple product Proof
$$(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{a}(\mathbf{b} \cdot \mathbf{c}) \tag{8}$$1.4 Quadruple products
(9) Lagrange identity Proof
$$(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}) \tag{9}$$(10) Squared magnitude of the cross product Proof
$$|\mathbf{a} \times \mathbf{b}|^2 = |\mathbf{a}|^2 |\mathbf{b}|^2 - (\mathbf{a} \cdot \mathbf{b})^2 \tag{10}$$(11) Vector quadruple product Proof
$$(\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d}) = \mathbf{c}\bigl(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{d})\bigr) - \mathbf{d}\bigl(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})\bigr) \tag{11}$$1.5 Jacobi identity
(12) Jacobi identity Proof
$$\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0} \tag{12}$$2. Gradient Identities
2.1 Linearity
(13) Linearity of the gradient Proof
$$\nabla(f + g) = \nabla f + \nabla g \tag{13}$$(14) Scalar multiple Proof
$$\nabla(cf) = c\,\nabla f \tag{14}$$2.2 Product rules
(15) Gradient of a product of scalars Proof
$$\nabla(fg) = f\,\nabla g + g\,\nabla f \tag{15}$$(16) Gradient of a dot product of vector fields Proof
$$\nabla(\mathbf{A} \cdot \mathbf{B}) = (\mathbf{B} \cdot \nabla)\mathbf{A} + (\mathbf{A} \cdot \nabla)\mathbf{B} + \mathbf{B} \times (\nabla \times \mathbf{A}) + \mathbf{A} \times (\nabla \times \mathbf{B}) \tag{16}$$2.3 Composite functions
(17) Chain rule Proof
$$\nabla f(g) = f'(g)\,\nabla g \tag{17}$$(18) Gradient of a power Proof
$$\nabla f^n = n f^{n-1}\,\nabla f \tag{18}$$2.4 Quotient rule
(19) Gradient of a scalar quotient Proof
$$\nabla\!\left(\dfrac{f}{g}\right) = \dfrac{g\,\nabla f - f\,\nabla g}{g^2} \tag{19}$$2.5 Special gradients
(20) Gradient of a Coulomb-type function Proof
$$\nabla\!\left(\dfrac{1}{|\mathbf{r} - \mathbf{r}'|}\right) = -\dfrac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \tag{20}$$3. Divergence Identities
3.1 Linearity
(21) Linearity of the divergence Proof
$$\nabla \cdot (\mathbf{A} + \mathbf{B}) = \nabla \cdot \mathbf{A} + \nabla \cdot \mathbf{B} \tag{21}$$(22) Scalar multiple Proof
$$\nabla \cdot (c\,\mathbf{A}) = c\,\nabla \cdot \mathbf{A} \tag{22}$$3.2 Product rules
(23) Divergence of a scalar times a vector field Proof
$$\nabla \cdot (f\mathbf{A}) = f(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f) \tag{23}$$(24) Divergence of a cross product Proof
$$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B}) \tag{24}$$3.3 Combinations with the gradient
(25) Divergence of a cross product of gradients Proof
$$\nabla \cdot (\nabla f \times \nabla g) = 0 \tag{25}$$(26) Differential form of Green's first identity Proof
$$\nabla \cdot (f\,\nabla g) = f\,\nabla^2 g + \nabla f \cdot \nabla g \tag{26}$$(27) Differential form of Green's second identity Proof
$$\nabla \cdot (f\,\nabla g - g\,\nabla f) = f\,\nabla^2 g - g\,\nabla^2 f \tag{27}$$(28) Divergence of a vector product (alternative form) Proof
$$\nabla \cdot (f\,\nabla g \times \nabla h) = \nabla f \cdot (\nabla g \times \nabla h) \tag{28}$$4. Curl Identities
4.1 Linearity
(29) Linearity of the curl Proof
$$\nabla \times (\mathbf{A} + \mathbf{B}) = \nabla \times \mathbf{A} + \nabla \times \mathbf{B} \tag{29}$$(30) Scalar multiple Proof
$$\nabla \times (c\,\mathbf{A}) = c\,(\nabla \times \mathbf{A}) \tag{30}$$4.2 Product rules
(31) Curl of a scalar times a vector field Proof
$$\nabla \times (f\mathbf{A}) = f(\nabla \times \mathbf{A}) + (\nabla f) \times \mathbf{A} \tag{31}$$(32) Curl of a cross product Proof
$$\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla)\mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B} \tag{32}$$4.3 Combinations with the gradient
(33) Curl of a cross product of gradients Proof
$$\nabla \times (\nabla f \times \nabla g) = \nabla f\,\nabla^2 g - \nabla g\,\nabla^2 f + (\nabla g \cdot \nabla)\nabla f - (\nabla f \cdot \nabla)\nabla g \tag{33}$$(34) Curl of a scalar times a gradient (alternative form) Proof
$$\nabla \times (f\,\nabla g) = \nabla f \times \nabla g \tag{34}$$(35) Skew-symmetry Proof
$$\nabla \times (f\,\nabla g) = -\nabla \times (g\,\nabla f) \tag{35}$$5. Second-Order Derivative Identities
5.1 The Laplacian
(36) Definition of the Laplacian Proof
$$\nabla \cdot (\nabla f) = \nabla^2 f = \Delta f \tag{36}$$(37) Laplacian in Cartesian coordinates Proof
$$\nabla^2 f = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2} \tag{37}$$5.2 Fundamental identities
(38) The curl of a gradient vanishes (condition for a conservative field) Proof
Holds for any $C^2$ scalar field $f$. This is the mathematical guarantee that a conservative force field admits a scalar potential.
(39) The divergence of a curl vanishes Proof
Holds for any $C^2$ vector field $\mathbf{A}$. This is the mathematical basis for the absence of magnetic monopoles, $\nabla \cdot \mathbf{B} = 0$.
(40) Curl of curl (vector Laplacian identity) Proof
$$\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A} \tag{40}$$5.3 Product rules for the Laplacian
(41) Laplacian of a product of scalars Proof
$$\nabla^2(fg) = f\,\nabla^2 g + 2(\nabla f \cdot \nabla g) + g\,\nabla^2 f \tag{41}$$(42) Laplacian of a vector field (componentwise) Proof
$$\nabla^2 \mathbf{A} = (\nabla^2 A_x,\; \nabla^2 A_y,\; \nabla^2 A_z) \tag{42}$$(43) Laplacian of a scalar times a vector field Proof
$$\nabla^2(f\mathbf{A}) = (\nabla^2 f)\mathbf{A} + 2(\nabla f \cdot \nabla)\mathbf{A} + f\,\nabla^2\mathbf{A} \tag{43}$$(44) Laplacian of a dot product Proof
$$\nabla^2(\mathbf{A} \cdot \mathbf{B}) = \mathbf{A} \cdot \nabla^2\mathbf{B} + \mathbf{B} \cdot \nabla^2\mathbf{A} + 2\displaystyle\sum_{i} (\nabla A_i \cdot \nabla B_i) \tag{44}$$6. Position Vector Identities
Let $\mathbf{r} = (x, y, z)$, $r = |\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$, and $\hat{\mathbf{r}} = \mathbf{r}/r$.
6.1 First-order derivatives
(45) Gradient of $r$ Proof
$$\nabla r = \hat{\mathbf{r}} = \dfrac{\mathbf{r}}{r} \tag{45}$$(46) Divergence of $\mathbf{r}$ Proof
$$\nabla \cdot \mathbf{r} = 3 \tag{46}$$(47) Curl of $\mathbf{r}$ Proof
$$\nabla \times \mathbf{r} = \mathbf{0} \tag{47}$$(48) Gradient of $1/r$ Proof
$$\nabla\!\left(\dfrac{1}{r}\right) = -\dfrac{\mathbf{r}}{r^3} \tag{48}$$(49) Gradient of $r^2$ Proof
$$\nabla r^2 = 2\mathbf{r} \tag{49}$$(50) Gradient of $r^n$ Proof
$$\nabla r^n = n\,r^{n-2}\,\mathbf{r} = n\,r^{n-1}\,\hat{\mathbf{r}} \tag{50}$$6.2 Divergence formulas
(51) Divergence of $r^n \mathbf{r}$ Proof
$$\nabla \cdot (r^n \mathbf{r}) = (n + 3)\,r^n \tag{51}$$(52) Divergence of $\mathbf{r}/r^3$ (delta function) Proof
$$\nabla \cdot \!\left(\dfrac{\mathbf{r}}{r^3}\right) = 4\pi\,\delta^3(\mathbf{r}) \tag{52}$$6.3 Laplacians
(53) Laplacian of $1/r$ Proof
$$\nabla^2\!\left(\dfrac{1}{r}\right) = -4\pi\,\delta^3(\mathbf{r}) \tag{53}$$(54) Laplacian of $r^n$ (for $r \neq 0$) Proof
$$\nabla^2 r^n = n(n+1)\,r^{n-2} \tag{54}$$(55) Laplacian of $\ln r$ (for $r \neq 0$) Proof
$$\nabla^2 \ln r = \dfrac{1}{r^2} \tag{55}$$6.4 (A·∇) acting on the position vector
(56) Proof
$$(\mathbf{A} \cdot \nabla)\mathbf{r} = \mathbf{A} \tag{56}$$(57) Proof
$$(\mathbf{A} \cdot \nabla)r = \dfrac{\mathbf{A} \cdot \mathbf{r}}{r} \tag{57}$$7. The $(\mathbf{A} \cdot \nabla)$ Operator
7.1 Definition
(58) Definition of $(\mathbf{A} \cdot \nabla)$ Proof
$$(\mathbf{A} \cdot \nabla)\mathbf{B} = \left(A_x \dfrac{\partial}{\partial x} + A_y \dfrac{\partial}{\partial y} + A_z \dfrac{\partial}{\partial z}\right)\mathbf{B} \tag{58}$$That is, componentwise, (59) Proof
$$\bigl[(\mathbf{A} \cdot \nabla)\mathbf{B}\bigr]_i = A_x \dfrac{\partial B_i}{\partial x} + A_y \dfrac{\partial B_i}{\partial y} + A_z \dfrac{\partial B_i}{\partial z} \tag{59}$$7.2 Basic identities
(60) Action on a scalar field Proof
$$(\mathbf{A} \cdot \nabla)f = \mathbf{A} \cdot \nabla f \tag{60}$$(61) Relation to the cross product with a curl Proof
$$\mathbf{A} \times (\nabla \times \mathbf{B}) = (\nabla \mathbf{B}) \cdot \mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B} \tag{61}$$Here $(\nabla \mathbf{B})$ denotes the Jacobian (dyadic) of the vector field $\mathbf{B}$, with components $(\nabla \mathbf{B})_{ij} = \partial B_j / \partial x_i$. In index notation, (62) Proof
$$\bigl[\mathbf{A} \times (\nabla \times \mathbf{B})\bigr]_i = \displaystyle\sum_j A_j \dfrac{\partial B_j}{\partial x_i} - \displaystyle\sum_j A_j \dfrac{\partial B_i}{\partial x_j} \tag{62}$$(63) Gradient of a dot product (alternative form of (16)) Proof
$$(\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} = \nabla(\mathbf{A} \cdot \mathbf{B}) - \mathbf{A} \times (\nabla \times \mathbf{B}) - \mathbf{B} \times (\nabla \times \mathbf{A}) \tag{63}$$7.3 Material derivative
In fluid mechanics, the material derivative along a velocity field $\mathbf{v}$ is expressed using the $(\mathbf{v} \cdot \nabla)$ operator.
(64) Material derivative of a scalar field Proof
$$\dfrac{Df}{Dt} = \dfrac{\partial f}{\partial t} + (\mathbf{v} \cdot \nabla)f \tag{64}$$(65) Material derivative of a vector field Proof
$$\dfrac{D\mathbf{A}}{Dt} = \dfrac{\partial \mathbf{A}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{A} \tag{65}$$7.4 Lamb's identity
The nonlinear term $(\mathbf{v} \cdot \nabla)\mathbf{v}$ in the Navier–Stokes equations can be rewritten as follows.
(66) Lamb's identity Proof
$$(\mathbf{v} \cdot \nabla)\mathbf{v} = \nabla\!\left(\dfrac{|\mathbf{v}|^2}{2}\right) - \mathbf{v} \times (\nabla \times \mathbf{v}) \tag{66}$$