Vector calculus identities

For a function ${\displaystyle f(x,y,z)}$ in three-dimensional Cartesian coordinate variables, the gradient is the vector field:

$${\displaystyle \operatorname {grad} (f)=\nabla f={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} }$$

where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables ${\displaystyle \psi (x_{1},\ldots ,x_{n})}$, also called a scalar field, the gradient is the vector field:

$${\displaystyle \nabla \psi ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\end{pmatrix}}\psi ={\frac {\partial \psi }{\partial x_{1}}}\mathbf {e} _{1}+\dots +{\frac {\partial \psi }{\partial x_{n}}}\mathbf {e} _{n}.}$$

where ${\displaystyle \mathbf {e} _{i}}$ are orthogonal unit vectors in arbitrary directions.

As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.

For a vector field ${\displaystyle \mathbf {A} =\left(A_{1},\ldots ,A_{n}\right)}$, also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix:

$${\displaystyle \mathbf {J} _{\mathbf {A} }=d\mathbf {A} =(\nabla \!\mathbf {A} )^{\textsf {T}}=\left({\frac {\partial A_{i}}{\partial x_{j}}}\right)_{\!ij}.}$$

For a tensor field ${\displaystyle \mathbf {T} }$ of any order k, the gradient ${\displaystyle \operatorname {grad} (\mathbf {T} )=d\mathbf {T} =(\nabla \mathbf {T} )^{\textsf {T}}}$ is a tensor field of order k + 1.

For a tensor field ${\displaystyle \mathbf {T} }$ of order k > 0, the tensor field ${\displaystyle \nabla \mathbf {T} }$ of order k + 1 is defined by the recursive relation

$${\displaystyle (\nabla \mathbf {T} )\cdot \mathbf {C} =\nabla (\mathbf {T} \cdot \mathbf {C} )}$$

where ${\displaystyle \mathbf {C} }$ is an arbitrary constant vector.

In Cartesian coordinates, the divergence of a continuously differentiable vector field ${\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }$ is the scalar-valued function:

$${\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\cdot {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.}$$

As the name implies the divergence is a measure of how much vectors are diverging.

The divergence of a tensor field ${\displaystyle \mathbf {T} }$ of non-zero order k is written as ${\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} }$, a contraction to a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity,

$${\displaystyle \nabla \cdot \left(\mathbf {A} \otimes \mathbf {T} \right)=\mathbf {T} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {T} }$$

where ${\displaystyle \mathbf {A} \cdot \nabla }$ is the directional derivative in the direction of ${\displaystyle \mathbf {A} }$ multiplied by its magnitude. Specifically, for the outer product of two vectors,

$${\displaystyle \nabla \cdot \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=\mathbf {B} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {B} .}$$

For a tensor field ${\displaystyle \mathbf {T} }$ of order k > 1, the tensor field ${\displaystyle \nabla \cdot \mathbf {T} }$ of order k − 1 is defined by the recursive relation

$${\displaystyle (\nabla \cdot \mathbf {T} )\cdot \mathbf {C} =\nabla \cdot (\mathbf {T} \cdot \mathbf {C} )}$$

where ${\displaystyle \mathbf {C} }$ is an arbitrary constant vector.

In Cartesian coordinates, for ${\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} }$ the curl is the vector field:

$${\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}\\[1em]&=\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} \end{aligned}}}$$

where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively.

As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.

In Einstein notation, the vector field ${\displaystyle \mathbf {F} ={\begin{pmatrix}F_{1},\ F_{2},\ F_{3}\end{pmatrix}}}$ has curl given by:

$${\displaystyle \nabla \times \mathbf {F} =\varepsilon ^{ijk}\mathbf {e} _{i}{\frac {\partial F_{k}}{\partial x_{j}}}}$$

where ${\displaystyle \varepsilon }$ = ±1 or 0 is the Levi-Civita parity symbol.

For a tensor field ${\displaystyle \mathbf {T} }$ of order k > 1, the tensor field ${\displaystyle \nabla \times \mathbf {T} }$ of order k is defined by the recursive relation

$${\displaystyle (\nabla \times \mathbf {T} )\cdot \mathbf {C} =\nabla \times (\mathbf {T} \cdot \mathbf {C} )}$$

where ${\displaystyle \mathbf {C} }$ is an arbitrary constant vector.

A tensor field of order greater than one may be decomposed into a sum of outer products, and then the following identity may be used:

$${\displaystyle \nabla \times \left(\mathbf {A} \otimes \mathbf {T} \right)=(\nabla \times \mathbf {A} )\otimes \mathbf {T} -\mathbf {A} \times (\nabla \mathbf {T} ).}$$

Specifically, for the outer product of two vectors,

$${\displaystyle \nabla \times \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=(\nabla \times \mathbf {A} )\mathbf {B} ^{\textsf {T}}-\mathbf {A} \times (\nabla \mathbf {B} ).}$$

In Cartesian coordinates, the Laplacian of a function ${\displaystyle f(x,y,z)}$ is

$${\displaystyle \Delta f=\nabla ^{2}\!f=\nabla \cdot (\nabla f)={\frac {\partial ^{2}\!f}{\partial x^{2}}}+{\frac {\partial ^{2}\!f}{\partial y^{2}}}+{\frac {\partial ^{2}\!f}{\partial z^{2}}}.}$$

The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.

When the Laplacian is equal to 0, the function is called a harmonic function. That is,

$${\displaystyle \Delta f=0.}$$

For a tensor field, ${\displaystyle \mathbf {T} }$, the Laplacian is generally written as:

$${\displaystyle \Delta \mathbf {T} =\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} }$$

and is a tensor field of the same order.

For a tensor field ${\displaystyle \mathbf {T} }$ of order k > 0, the tensor field ${\displaystyle \nabla ^{2}\mathbf {T} }$ of order k is defined by the recursive relation

$${\displaystyle \left(\nabla ^{2}\mathbf {T} \right)\cdot \mathbf {C} =\nabla ^{2}(\mathbf {T} \cdot \mathbf {C} )}$$

where ${\displaystyle \mathbf {C} }$ is an arbitrary constant vector.

In Feynman subscript notation,

$${\displaystyle \nabla _{\mathbf {B} }\!\left(\mathbf {A{\cdot }B} \right)=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} }$$

where the notation ∇_B means the subscripted gradient operates on only the factor B.[1][2]

Less general but similar is the Hestenes overdot notation in geometric algebra.[3] The above identity is then expressed as:

$${\displaystyle {\dot {\nabla }}\left(\mathbf {A} {\cdot }{\dot {\mathbf {B} }}\right)=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} }$$

where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant.

For the remainder of this article, Feynman subscript notation will be used where appropriate.

${\displaystyle {\begin{aligned}\nabla (\psi +\phi )&=\nabla \psi +\nabla \phi \\\nabla (\mathbf {A} +\mathbf {B} )&=\nabla \mathbf {A} +\nabla \mathbf {B} \\\nabla \cdot (\mathbf {A} +\mathbf {B} )&=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} \\\nabla \times (\mathbf {A} +\mathbf {B} )&=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} \end{aligned}}}$

${\displaystyle {\begin{aligned}(\mathbf {A} \cdot \nabla )\psi &=\mathbf {A} \cdot (\nabla \psi )\\(\mathbf {A} \cdot \nabla )\mathbf {B} &=\mathbf {A} \cdot (\nabla \mathbf {B} )\\(\mathbf {A} \times \nabla )\psi &=\mathbf {A} \times (\nabla \psi )\\(\mathbf {A} \times \nabla )\mathbf {B} &=\mathbf {A} \times (\nabla \mathbf {B} )\end{aligned}}}$

We have the following generalizations of the product rule in single variable calculus.

${\displaystyle {\begin{aligned}\nabla (\psi \phi )&=\phi \,\nabla \psi +\psi \,\nabla \phi \\\nabla (\psi \mathbf {A} )&=(\nabla \psi )\mathbf {A} ^{\textsf {T}}+\psi \nabla \mathbf {A} \ =\ \nabla \psi \otimes \mathbf {A} +\psi \,\nabla \mathbf {A} \\\nabla \cdot (\psi \mathbf {A} )&=\psi \,\nabla {\cdot }\mathbf {A} +(\nabla \psi )\,{\cdot }\mathbf {A} \\\nabla {\times }(\psi \mathbf {A} )&=\psi \,\nabla {\times }\mathbf {A} +(\nabla \psi ){\times }\mathbf {A} \\\nabla ^{2}(\psi \phi )&=\psi \,\nabla ^{2\!}\phi +2\,\nabla \!\psi \cdot \!\nabla \phi +\phi \,\nabla ^{2\!}\psi \end{aligned}}}$

${\displaystyle {\begin{aligned}\nabla \left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla \psi -\psi \,\nabla \phi }{\phi ^{2}}}\\[1em]\nabla \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla \mathbf {A} -\nabla \phi \otimes \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \cdot \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\cdot }\mathbf {A} -\nabla \!\phi \cdot \mathbf {A} }{\phi ^{2}}}\\[1em]\nabla \times \left({\frac {\mathbf {A} }{\phi }}\right)&={\frac {\phi \,\nabla {\times }\mathbf {A} -\nabla \!\phi \,{\times }\,\mathbf {A} }{\phi ^{2}}}\\[1em]\nabla ^{2}\left({\frac {\psi }{\phi }}\right)&={\frac {\phi \,\nabla ^{2\!}\psi -2\,\phi \,\nabla \!\left({\frac {\psi }{\phi }}\right)\cdot \!\nabla \phi -\psi \,\nabla ^{2\!}\phi }{\phi ^{2}}}\end{aligned}}}$

Let ${\displaystyle f(x)}$ be a one-variable function from scalars to scalars, ${\displaystyle \mathbf {r} (t)=(x_{1}(t),\ldots ,x_{n}(t))}$ a parametrized curve, ${\displaystyle \phi \!:\mathbb {R} ^{n}\to \mathbb {R} }$ a function from vectors to scalars, and ${\displaystyle \mathbf {A} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ a vector field. We have the following special cases of the multi-variable chain rule.

${\displaystyle {\begin{aligned}\nabla (f\circ \phi )&=\left(f'\circ \phi \right)\nabla \phi \\(\mathbf {r} \circ f)'&=(\mathbf {r} '\circ f)f'\\(\phi \circ \mathbf {r} )'&=(\nabla \phi \circ \mathbf {r} )\cdot \mathbf {r} '\\(\mathbf {A} \circ \mathbf {r} )'&=\mathbf {r} '\cdot (\nabla \mathbf {A} \circ \mathbf {r} )\\\nabla (\phi \circ \mathbf {A} )&=(\nabla \mathbf {A} )\cdot (\nabla \phi \circ \mathbf {A} )\\\nabla \cdot (\mathbf {r} \circ \phi )&=\nabla \phi \cdot (\mathbf {r} '\circ \phi )\\\nabla \times (\mathbf {r} \circ \phi )&=\nabla \phi \times (\mathbf {r} '\circ \phi )\end{aligned}}}$

For a vector transformation ${\displaystyle \mathbf {x} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ we have:

${\displaystyle \nabla \cdot (\mathbf {A} \circ \mathbf {x} )=\mathrm {tr} \left((\nabla \mathbf {x} )\cdot (\nabla \mathbf {A} \circ \mathbf {x} )\right)}$

Here we take the trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.

${\displaystyle {\begin{aligned}\nabla (\mathbf {A} \cdot \mathbf {B} )&\ =\ (\mathbf {A} \cdot \nabla )\mathbf {B} \,+\,(\mathbf {B} \cdot \nabla )\mathbf {A} \,+\,\mathbf {A} {\times }(\nabla {\times }\mathbf {B} )\,+\,\mathbf {B} {\times }(\nabla {\times }\mathbf {A} )\\&\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {B} }+\mathbf {B} \cdot \mathbf {J} _{\mathbf {A} }\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,+\,(\nabla \mathbf {A} )\cdot \mathbf {B} \end{aligned}}}$

where ${\displaystyle \mathbf {J} _{\mathbf {A} }=(\nabla \!\mathbf {A} )^{\textsf {T}}=(\partial A_{i}/\partial x_{j})_{ij}}$ denotes the Jacobian matrix of the vector field ${\displaystyle \mathbf {A} =(A_{1},\ldots ,A_{n})}$.

Alternatively, using Feynman subscript notation,

${\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=\nabla _{\mathbf {A} }(\mathbf {A} \cdot \mathbf {B} )+\nabla _{\mathbf {B} }(\mathbf {A} \cdot \mathbf {B} )\ .}$

See these notes.[4]

As a special case, when A = B,

${\displaystyle {\tfrac {1}{2}}\nabla \left(\mathbf {A} \cdot \mathbf {A} \right)\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {A} }\ =\ (\nabla \mathbf {A} )\cdot \mathbf {A} \ =\ (\mathbf {A} {\cdot }\nabla )\mathbf {A} \,+\,\mathbf {A} {\times }(\nabla {\times }\mathbf {A} )\ =\ A\nabla A.}$

The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection, which differentiates a vector field to give a vector-valued 1-form.

${\displaystyle {\begin{aligned}\nabla \cdot (\mathbf {A} \times \mathbf {B} )&\ =\ (\nabla {\times }\mathbf {A} )\cdot \mathbf {B} \,-\,\mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\\[5pt]\nabla \times (\mathbf {A} \times \mathbf {B} )&\ =\ \mathbf {A} (\nabla {\cdot }\mathbf {B} )\,-\,\mathbf {B} (\nabla {\cdot }\mathbf {A} )\,+\,(\mathbf {B} {\cdot }\nabla )\mathbf {A} \,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ \mathbf {A} (\nabla {\cdot }\mathbf {B} )\,+\,(\mathbf {B} {\cdot }\nabla )\mathbf {A} \,-\,(\mathbf {B} (\nabla {\cdot }\mathbf {A} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} )\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\right)\,-\,\nabla {\cdot }\left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[2pt]&\ =\ \nabla {\cdot }\left(\mathbf {B} \mathbf {A} ^{\textsf {T}}\,-\,\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)\\[5pt]\mathbf {A} \times (\nabla \times \mathbf {B} )&\ =\ \nabla _{\mathbf {B} }(\mathbf {A} {\cdot }\mathbf {B} )\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ \mathbf {A} \cdot \mathbf {J} _{\mathbf {B} }\,-\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \\[2pt]&\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} \cdot (\nabla \mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \cdot (\mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}})\\[5pt](\mathbf {A} \times \nabla )\times \mathbf {B} &\ =\ (\nabla \mathbf {B} )\cdot \mathbf {A} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[2pt]&\ =\ \mathbf {A} \times (\nabla \times \mathbf {B} )\,+\,(\mathbf {A} {\cdot }\nabla )\mathbf {B} \,-\,\mathbf {A} (\nabla {\cdot }\mathbf {B} )\\[5pt](\mathbf {A} \times \nabla )\cdot \mathbf {B} &\ =\ \mathbf {A} \cdot (\nabla {\times }\mathbf {B} )\end{aligned}}}$

Note that the matrix ${\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}}}$ is antisymmetric.

The divergence of the curl of any continuously twice-differentiable vector field A is always zero:

$${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$$

This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

The Laplacian of a scalar field is the divergence of its gradient:

$${\displaystyle \Delta \psi =\nabla ^{2}\psi =\nabla \cdot (\nabla \psi )}$$

The result is a scalar quantity.

Divergence of a vector field A is a scalar, and you cannot take the divergence of a scalar quantity. Therefore:

$${\displaystyle \nabla \cdot (\nabla \cdot \mathbf {A} ){\text{ is undefined}}}$$

The curl of the gradient of any continuously twice-differentiable scalar field ${\displaystyle \varphi }$ (i.e., differentiability class ${\displaystyle C^{2}}$) is always the zero vector:

$${\displaystyle \nabla \times (\nabla \varphi )=\mathbf {0} }$$

It can be easily proved by expressing ${\displaystyle \nabla \times (\nabla \varphi )}$ in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex.

$${\displaystyle \nabla \times \left(\nabla \times \mathbf {A} \right)\ =\ \nabla (\nabla {\cdot }\mathbf {A} )\,-\,\nabla ^{2\!}\mathbf {A} }$$

Here ∇² is the vector Laplacian operating on the vector field A.

The divergence of a vector field A is a scalar, and you cannot take curl of a scalar quantity. Therefore

$${\displaystyle \nabla \times (\nabla \cdot \mathbf {A} ){\text{ is undefined}}}$$

${\displaystyle {\begin{aligned}(\nabla \cdot \nabla )\psi &=\nabla \cdot (\nabla \psi )=\nabla ^{2}\psi \\(\nabla \cdot \nabla )\mathbf {A} &=\nabla \cdot (\nabla \mathbf {A} )=\nabla ^{2}\mathbf {A} \\(\nabla \times \nabla )\psi &=\nabla \times (\nabla \psi )=\mathbf {0} \\(\nabla \times \nabla )\mathbf {A} &=\nabla \times (\nabla \mathbf {A} )=\mathbf {0} \end{aligned}}}$

DCG chart: Some rules for second derivatives.

The figure to the right is a mnemonic for some of these identities. The abbreviations used are:

• D: divergence,
• C: curl,
• G: gradient,
• L: Laplacian,
• CC: curl of curl.

Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.

• ${\displaystyle \nabla (\psi +\phi )=\nabla \psi +\nabla \phi }$
• ${\displaystyle \nabla (\psi \phi )=\phi \nabla \psi +\psi \nabla \phi }$
• ${\displaystyle \nabla (\psi \mathbf {A} )=\nabla \psi \otimes \mathbf {A} +\psi \nabla \mathbf {A} }$
• ${\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} )}$

• ${\displaystyle \nabla \cdot (\mathbf {A} +\mathbf {B} )=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} }$
• ${\displaystyle \nabla \cdot \left(\psi \mathbf {A} \right)=\psi \nabla \cdot \mathbf {A} +\mathbf {A} \cdot \nabla \psi }$
• ${\displaystyle \nabla \cdot \left(\mathbf {A} \times \mathbf {B} \right)=(\nabla \times \mathbf {A} )\cdot \mathbf {B} -(\nabla \times \mathbf {B} )\cdot \mathbf {A} }$

• ${\displaystyle \nabla \times (\mathbf {A} +\mathbf {B} )=\nabla \times \mathbf {A} +\nabla \times \mathbf {B} }$
• ${\displaystyle \nabla \times \left(\psi \mathbf {A} \right)=\psi \,(\nabla \times \mathbf {A} )-(\mathbf {A} \times \nabla )\psi =\psi \,(\nabla \times \mathbf {A} )+(\nabla \psi )\times \mathbf {A} }$
• ${\displaystyle \nabla \times \left(\psi \nabla \phi \right)=\nabla \psi \times \nabla \phi }$
• ${\displaystyle \nabla \times \left(\mathbf {A} \times \mathbf {B} \right)=\mathbf {A} \left(\nabla \cdot \mathbf {B} \right)-\mathbf {B} \left(\nabla \cdot \mathbf {A} \right)+\left(\mathbf {B} \cdot \nabla \right)\mathbf {A} -\left(\mathbf {A} \cdot \nabla \right)\mathbf {B} }$[5]

• ${\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} ={\frac {1}{2}}{\bigg [}\nabla (\mathbf {A} \cdot \mathbf {B} )-\nabla \times (\mathbf {A} \times \mathbf {B} )-\mathbf {B} \times (\nabla \times \mathbf {A} )-\mathbf {A} \times (\nabla \times \mathbf {B} )-\mathbf {B} (\nabla \cdot \mathbf {A} )+\mathbf {A} (\nabla \cdot \mathbf {B} ){\bigg ]}}$[6]

• ${\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} ={\frac {1}{2}}\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2}}\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }$

• ${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$
• ${\displaystyle \nabla \times (\nabla \psi )=\mathbf {0} }$
• ${\displaystyle \nabla \cdot (\nabla \psi )=\nabla ^{2}\psi }$ (scalar Laplacian)
• ${\displaystyle \nabla \left(\nabla \cdot \mathbf {A} \right)-\nabla \times \left(\nabla \times \mathbf {A} \right)=\nabla ^{2}\mathbf {A} }$ (vector Laplacian)
• ${\displaystyle \nabla \cdot (\phi \nabla \psi )=\phi \nabla ^{2}\psi +\nabla \phi \cdot \nabla \psi }$
• ${\displaystyle \psi \nabla ^{2}\phi -\phi \nabla ^{2}\psi =\nabla \cdot \left(\psi \nabla \phi -\phi \nabla \psi \right)}$
• ${\displaystyle \nabla ^{2}(\phi \psi )=\phi \nabla ^{2}\psi +2(\nabla \phi )\cdot (\nabla \psi )+\left(\nabla ^{2}\phi \right)\psi }$
• ${\displaystyle \nabla ^{2}(\psi \mathbf {A} )=\mathbf {A} \nabla ^{2}\psi +2(\nabla \psi \cdot \nabla )\mathbf {A} +\psi \nabla ^{2}\mathbf {A} }$
• ${\displaystyle \nabla ^{2}(\mathbf {A} \cdot \mathbf {B} )=\mathbf {A} \cdot \nabla ^{2}\mathbf {B} -\mathbf {B} \cdot \nabla ^{2}\!\mathbf {A} +2\nabla \cdot ((\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {B} \times (\nabla \times \mathbf {A} ))}$ (Green's vector identity)

• ${\displaystyle \nabla ^{2}(\nabla \psi )=\nabla (\nabla \cdot (\nabla \psi ))=\nabla \left(\nabla ^{2}\psi \right)}$
• ${\displaystyle \nabla ^{2}(\nabla \cdot \mathbf {A} )=\nabla \cdot (\nabla (\nabla \cdot \mathbf {A} ))=\nabla \cdot \left(\nabla ^{2}\mathbf {A} \right)}$
• ${\displaystyle \nabla ^{2}(\nabla \times \mathbf {A} )=-\nabla \times (\nabla \times (\nabla \times \mathbf {A} ))=\nabla \times \left(\nabla ^{2}\mathbf {A} \right)}$

Below, the curly symbol ∂ means "boundary of" a surface or solid.

In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂V (a closed surface):

• ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \psi \,d\mathbf {S} \ =\ \iiint _{V}\nabla \psi \,dV}$
• ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \mathbf {A} \cdot d\mathbf {S} \ =\ \iiint _{V}\nabla \cdot \mathbf {A} \,dV}$ (divergence theorem)
• ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \mathbf {A} \times d\mathbf {S} \ =\ -\iiint _{V}\nabla \times \mathbf {A} \,dV}$
• ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \psi \nabla \!\varphi \cdot d\mathbf {S} \ =\ \iiint _{V}\left(\psi \nabla ^{2}\!\varphi +\nabla \!\varphi \cdot \nabla \!\psi \right)\,dV}$ (Green's first identity)
• ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \left(\psi \nabla \!\varphi -\varphi \nabla \!\psi \right)\cdot d\mathbf {S} \ =\ }$ ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \left(\psi {\frac {\partial \varphi }{\partial n}}-\varphi {\frac {\partial \psi }{\partial n}}\right)dS}$ ${\displaystyle \displaystyle \ =\ \iiint _{V}\left(\psi \nabla ^{2}\!\varphi -\varphi \nabla ^{2}\!\psi \right)\,dV}$ (Green's second identity)
• ${\displaystyle \iiint _{V}\mathbf {A} \cdot \nabla \psi \,dV\ =\ }$ ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \psi \mathbf {A} \cdot d\mathbf {S} -\iiint _{V}\psi \nabla \cdot \mathbf {A} \,dV}$ (integration by parts)
• ${\displaystyle \iiint _{V}\psi \nabla \cdot \mathbf {A} \,dV\ =\ }$ ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \psi \mathbf {A} \cdot d\mathbf {S} -\iiint _{V}\mathbf {A} \cdot \nabla \psi \,dV}$ (integration by parts)
• ${\displaystyle \iiint _{V}\mathbf {A} \cdot \left(\nabla \times \mathbf {B} \right)\,dV\ =\ -}$ ${\displaystyle \scriptstyle \partial V}$ ${\displaystyle \left(\mathbf {A} \times \mathbf {B} \right)\cdot d\mathbf {S} +\iiint _{V}\left(\nabla \times \mathbf {A} \right)\cdot \mathbf {B} \,dV}$ (integration by parts)

In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂S (a closed curve):

• ${\displaystyle \oint _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}\ =\ \iint _{S}\left(\nabla \times \mathbf {A} \right)\cdot d\mathbf {S} }$ (Stokes' theorem)
• ${\displaystyle \oint _{\partial S}\psi \,d{\boldsymbol {\ell }}\ =\ -\iint _{S}\nabla \psi \times d\mathbf {S} }$
• ${\displaystyle \oint _{\partial S}\mathbf {A} \times d{\boldsymbol {\ell }}\ =\ -\iint _{S}\left(\nabla \mathbf {A} -(\nabla \cdot \mathbf {A} )\mathbf {1} \right)\cdot d\mathbf {S} \ =\ -\iint _{S}\left(d\mathbf {S} \times \nabla \right)\times \mathbf {A} }$

Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral):

${\displaystyle {\scriptstyle \partial S}}$ ${\displaystyle \mathbf {A} \cdot d{\boldsymbol {\ell }}=-}$ ${\displaystyle {\scriptstyle \partial S}}$ ${\displaystyle \mathbf {A} \cdot d{\boldsymbol {\ell }}.}$

In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points ${\displaystyle \mathbf {q} -\mathbf {p} =\partial P}$ and integration along P is from ${\displaystyle \mathbf {p} }$ to ${\displaystyle \mathbf {q} }$:

• ${\displaystyle \psi |_{\partial P}=\psi (\mathbf {q} )-\psi (\mathbf {p} )=\int _{P}\nabla \psi \cdot d{\boldsymbol {\ell }}}$ (gradient theorem)
• ${\displaystyle \mathbf {A} |_{\partial P}=\mathbf {A} (\mathbf {q} )-\mathbf {A} (\mathbf {p} )=\int _{P}\left(d{\boldsymbol {\ell }}\cdot \nabla \right)\mathbf {A} }$