Vector Calculus Identities
A Complete Reference
This page collects the standard identities of vector calculus in three-dimensional Euclidean space $\mathbb{R}^3$. Notation: $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are constant vectors in $\mathbb{R}^3$ (not fields), $c$ is a constant scalar independent of position (a different symbol from the bold $\mathbf{c}$), $f, g, h$ are scalar fields, $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$ are vector fields, and $n$ is a real number.
In every identity involving derivatives the fields are assumed to be as smooth as the identity requires: $C^1$ where only first derivatives appear, and $C^2$ for the fields that are differentiated twice or whose mixed partial derivatives are interchanged ((25)–(28) and (33)–(44)). The identities themselves are coordinate-free, but the component forms (37) and (42)–(44) hold only in Cartesian coordinates. For the position-vector identities (§6) the origin needs separate treatment, so classical identities valid for $r \neq 0$ are distinguished from identities that hold in the distributional sense. The precise conditions for each identity are listed under "Conditions" on the linked proof page.
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 ($c$ is a scalar) 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 (also called the Binet–Cauchy 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 (the special case $\mathbf{c} = \mathbf{a}$, $\mathbf{d} = \mathbf{b}$ of (9)) 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
$$\nabla f(g) = f'(g)\,\nabla g \tag{17}$$In (17), $f$ is a function of one variable, $f\colon \mathbb{R} \to \mathbb{R}$ (and $f'$ is its derivative), while $g$ is a scalar field; this $f$ plays a different role from the scalar field $f$ in the other identities on this page.
(18) Gradient of a power ($n$ real) Proof
$$\nabla f^n = n f^{n-1}\,\nabla f \tag{18}$$For a positive integer $n$ the identity holds as stated. For negative or non-integer $n$ it is to be used only where $f \neq 0$, that is, on the region where $f^n$ is defined and differentiable.
2.4 Quotient rule
(19) Gradient of a scalar quotient ($g \neq 0$) Proof
$$\nabla\!\left(\dfrac{f}{g}\right) = \dfrac{g\,\nabla f - f\,\nabla g}{g^2} \tag{19}$$2.5 Representative Gradient Formulas
(20) Gradient of a Coulomb-type function Proof
$$\nabla\!\left(\dfrac{1}{|\mathbf{r} - \mathbf{r}'|}\right) = -\dfrac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \tag{20}$$Here $\nabla$ is the gradient with respect to $\mathbf{r}$, with $\mathbf{r}'$ held fixed (differentiating with respect to $\mathbf{r}'$ instead reverses the sign). Valid for $\mathbf{r} \neq \mathbf{r}'$.
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 (a gradient field is irrotational) Proof
Holds for any $C^2$ scalar field $f$. The statement runs in one direction only: a field built from a potential (a gradient field) is necessarily irrotational, which is why a conservative force field is curl-free. To conclude conversely that $\nabla \times \mathbf{F} = \mathbf{0}$ implies $\mathbf{F} = \nabla f$ for some potential $f$, the domain must be simply connected (Poincaré's lemma; on a domain that is not simply connected there are counterexamples, see Gradient, Divergence, Curl).
(39) The divergence of a curl vanishes (a curl field is solenoidal) Proof
Holds for any $C^2$ vector field $\mathbf{A}$. Again the statement runs in one direction: a field expressed as a curl is divergence-free. When the magnetic field is written as $\mathbf{B} = \nabla \times \mathbf{A}$, it follows that $\nabla \cdot \mathbf{B} = 0$ (the absence of magnetic monopoles itself is one of Maxwell's equations, not a consequence of this identity). To conclude conversely that $\nabla \cdot \mathbf{B} = 0$ implies the existence of a vector potential, conditions on the domain are required.
(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, Cartesian coordinates) Proof
$$\nabla^2 \mathbf{A} = (\nabla^2 A_x,\; \nabla^2 A_y,\; \nabla^2 A_z) \tag{42}$$Valid in Cartesian coordinates only. In curvilinear coordinates (cylindrical, spherical, ...) the basis vectors depend on position, and $\nabla^2 \mathbf{A}$ is no longer the componentwise scalar Laplacian (see Coordinate Systems). In a general coordinate system, (40) is taken as the definition of $\nabla^2 \mathbf{A}$. Identities (43) and (44) are likewise Cartesian-coordinate formulas.
(43) Laplacian of a scalar times a vector field (Cartesian coordinates) 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 (Cartesian coordinates) 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, for $r \neq 0$, $\hat{\mathbf{r}} = \mathbf{r}/r$. Expressions containing $\hat{\mathbf{r}}$ or $1/r$ are undefined at the origin; the range of validity of each identity is stated in its label.
6.1 First-order derivatives
(45) Gradient of $r$ ($r \neq 0$) Proof
$$\nabla r = \hat{\mathbf{r}} = \dfrac{\mathbf{r}}{r} \tag{45}$$(46) Divergence of $\mathbf{r}$ (valid for all $\mathbf{r}$) Proof
$$\nabla \cdot \mathbf{r} = 3 \tag{46}$$(47) Curl of $\mathbf{r}$ (valid for all $\mathbf{r}$) Proof
$$\nabla \times \mathbf{r} = \mathbf{0} \tag{47}$$(48) Gradient of $1/r$ ($r \neq 0$) Proof
$$\nabla\!\left(\dfrac{1}{r}\right) = -\dfrac{\mathbf{r}}{r^3} \tag{48}$$(49) Gradient of $r^2$ (valid for all $\mathbf{r}$) Proof
$$\nabla r^2 = 2\mathbf{r} \tag{49}$$(50) Gradient of $r^n$ ($r \neq 0$; also valid at $r = 0$ when $n \geq 2$) 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 $r^n \mathbf{r}$ ($r \neq 0$; also valid at $r = 0$ when $n \geq 0$) Proof
$$\nabla \cdot (r^n \mathbf{r}) = (n + 3)\,r^n \tag{51}$$Setting $n = -3$ in (51) gives $0$ on the right-hand side, but that is a statement for $r \neq 0$ only. Including the origin and reading the equation in the distributional sense, the following delta-function term appears.
(52) Divergence of $\mathbf{r}/r^3$ (delta function; holds in the distributional sense) Proof
$$\nabla \cdot \!\left(\dfrac{\mathbf{r}}{r^3}\right) = 4\pi\,\delta^3(\mathbf{r}) \tag{52}$$6.3 Laplacians
(53) Laplacian of $1/r$ (holds in the distributional sense) Proof
$$\nabla^2\!\left(\dfrac{1}{r}\right) = -4\pi\,\delta^3(\mathbf{r}) \tag{53}$$Identities (52) and (53) hold in the sense of distributions: the claim is that both sides give the same value when multiplied by an arbitrary test function $\varphi$ and integrated over all space. Here $\delta^3(\mathbf{r})$ is the three-dimensional Dirac delta, defined by $\int \varphi(\mathbf{r})\,\delta^3(\mathbf{r})\,dV = \varphi(\mathbf{0})$. For $r \neq 0$ both sides vanish, so as pointwise identities $\nabla \cdot (\mathbf{r}/r^3) = 0$ and $\nabla^2(1/r) = 0$.
(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
$(\mathbf{A} \cdot \nabla)$ is a directional-derivative operator that acts only on the object to its right; the coefficient field $\mathbf{A}$ itself is not differentiated (even when $\mathbf{A}$ depends on position).
(56) Action on the position vector (valid for all $\mathbf{r}$) Proof
$$(\mathbf{A} \cdot \nabla)\mathbf{r} = \mathbf{A} \tag{56}$$(57) Action on $r$ ($r \neq 0$) 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
$$(\mathbf{A} \cdot \nabla)\mathbf{B} = \left(A_x \dfrac{\partial}{\partial x} + A_y \dfrac{\partial}{\partial y} + A_z \dfrac{\partial}{\partial z}\right)\mathbf{B} \tag{58}$$(59) Componentwise form of (58) ($i = x, y, z$) 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
$$(\mathbf{A} \cdot \nabla)f = \mathbf{A} \cdot \nabla f \tag{60}$$In what follows, $(\nabla \mathbf{B})$ denotes the Jacobian (dyadic) of the vector field $\mathbf{B}$. On this page its components are $(\nabla \mathbf{B})_{ij} = \partial B_j / \partial x_i$, and the product with a vector on the right is $[(\nabla \mathbf{B}) \cdot \mathbf{A}]_i = \sum_j A_j\,\partial B_j / \partial x_i$. In the convention that writes the usual Jacobian matrix as $(J_{\mathbf{B}})_{ij} = \partial B_i / \partial x_j$, $(\nabla \mathbf{B})$ is its transpose $J_{\mathbf{B}}^{\top}$. Index conventions vary between references, so watch for this transpose when comparing (61) and (62) with other sources.
(61) Relation to the cross product with a curl Proof
$$\mathbf{A} \times (\nabla \times \mathbf{B}) = (\nabla \mathbf{B}) \cdot \mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B} \tag{61}$$(62) Componentwise form of (61) 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 (the rate of change seen by an observer moving with a fluid particle) is the sum of the local time derivative $\partial/\partial t$ and the advective term $(\mathbf{v} \cdot \nabla)$ along the velocity field $\mathbf{v}$.
(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
$$(\mathbf{v} \cdot \nabla)\mathbf{v} = \nabla\!\left(\dfrac{|\mathbf{v}|^2}{2}\right) - \mathbf{v} \times (\nabla \times \mathbf{v}) \tag{66}$$Summary
The most frequently used identities on this page are collected below. Here $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are constant vectors, $f, g$ are scalar fields, and $\mathbf{A}, \mathbf{B}$ are vector fields.
| Category | Identity |
|---|---|
| Vector triple product (BAC-CAB) | $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})$ |
| Gradient of a product | $\nabla(fg) = f\,\nabla g + g\,\nabla f$ |
| Divergence of a product | $\nabla \cdot (f\mathbf{A}) = f(\nabla \cdot \mathbf{A}) + \mathbf{A} \cdot (\nabla f)$ |
| Divergence of a cross product | $\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$ |
| Curl of a cross product | $\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla)\mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B}$ |
| Identically zero (1) | $\nabla \times (\nabla f) = \mathbf{0}$ (a gradient field is irrotational) |
| Identically zero (2) | $\nabla \cdot (\nabla \times \mathbf{A}) = 0$ (a curl field is solenoidal) |
| Laplacian | $\nabla \cdot (\nabla f) = \nabla^2 f = \Delta f$ |
| Curl of a curl | $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ |
| Position vector | $\nabla r = \dfrac{\mathbf{r}}{r}$, $\nabla \cdot \mathbf{r} = 3$, $\nabla \times \mathbf{r} = \mathbf{0}$, $\nabla\!\left(\dfrac{1}{r}\right) = -\dfrac{\mathbf{r}}{r^3}$ |
| Lamb's identity | $(\mathbf{v} \cdot \nabla)\mathbf{v} = \nabla\!\left(\dfrac{|\mathbf{v}|^2}{2}\right) - \mathbf{v} \times (\nabla \times \mathbf{v})$ |
Points to remember
- The two identically zero relations appear constantly. $\nabla \times (\nabla f) = \mathbf{0}$ says that a gradient field is irrotational, and $\nabla \cdot (\nabla \times \mathbf{A}) = 0$ says that a field expressed as a curl is divergence-free; the latter corresponds to $\nabla \cdot \mathbf{B} = 0$ whenever the magnetic field is written as $\mathbf{B} = \nabla \times \mathbf{A}$. To conclude conversely that a zero condition implies the existence of a potential, conditions on the domain (such as simple connectivity) are required.
- These second-derivative identities hold whenever the fields are of class $C^2$, so that mixed partial derivatives commute. $C^2$ is a sufficient condition, not a necessary one; but drop the smoothness assumption and this argument no longer guarantees them (there are functions whose mixed partial derivatives disagree). For fields with singularities, distinguish classical identities valid for $r \neq 0$ from identities in the distributional sense.
- The curl-of-a-curl identity $\nabla \times (\nabla \times \mathbf{A}) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^2 \mathbf{A}$ is the key step in deriving the wave equation from Maxwell's equations.
- The position-vector formulas appear again and again in potentials of point sources (gravitational and electrostatic fields).
Frequently Asked Questions
What are the most important identities in vector calculus?
The two most important are $\nabla \times (\nabla f) = \mathbf{0}$ and $\nabla \cdot (\nabla \times \mathbf{A}) = 0$. The first says that a gradient field is irrotational; the second says that a curl field is divergence-free. To conclude conversely that a curl-free field has a potential, extra conditions on the domain (such as simple connectivity) are required (see (38) and (39)).
What is the BAC-CAB rule?
The BAC-CAB rule is $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})$. It expands the vector triple product as a linear combination of dot products and appears constantly in electromagnetism and fluid mechanics (see (7)).