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The following are importantidentities involving derivatives and integrals invector calculus.

For a function${\displaystyle f(x,y,z)}$ in three-dimensionalCartesian coordinate variables, the gradient is the vector field:$${\displaystyle \mathrm{grad}(f)=\mathrm{\nabla }f=\left(\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x},\frac{\mathrm{\partial }}{\mathrm{\partial }y},\frac{\mathrm{\partial }}{\mathrm{\partial }z}}\end{array}\right)f=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\mathbf{i}+\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\mathbf{j}+\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\mathbf{k}}$$wherei,j,k are thestandardunit vectors for thex,y,z-axes. More generally, for a function ofn variables${\displaystyle \psi (x_1,\dots ,x_n)}$, also called ascalar field, the gradient is thevector field:$${\displaystyle \mathrm{\nabla }\psi =\left(\begin{array}{c}{\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\dots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_n}}\end{array}\right)\psi =\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{x}_1}{\mathbf{e}}_1+\cdots +\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{x}_n}{\mathbf{e}}_n}$$where${\displaystyle {\mathbf{e}}_i(i=1,2,...,n)}$ are mutually orthogonal unit vectors.

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${\displaystyle \mathbf{A}=\left(A_1,\dots ,A_n\right)}$, also called a tensor field of order 1, the gradient ortotal derivative is then × nJacobian matrix:^[1]$${\displaystyle {\mathbf{J}}_{\mathbf{A}}=d\mathbf{A}=(\mathrm{\nabla }\mathbf{A}{)}^{\mathsf{\text{T}}}={\left(\frac{\mathrm{\partial }{A}_i}{\mathrm{\partial }{x}_j}\right)}_{ij}.}$$

For atensor field${\displaystyle \mathbf{T}}$ of any orderk, the gradient${\displaystyle \mathrm{grad}(\mathbf{T})=d\mathbf{T}=(\mathrm{\nabla }\mathbf{T}{)}^{\mathsf{\text{T}}}}$ is a tensor field of orderk + 1.

For a tensor field${\displaystyle \mathbf{T}}$ of orderk > 0, the tensor field${\displaystyle \mathrm{\nabla }\mathbf{T}}$ of orderk + 1 is defined by the recursive relation$${\displaystyle (\mathrm{\nabla }\mathbf{T})\cdot \mathbf{C}=\mathrm{\nabla }(\mathbf{T}\cdot \mathbf{C})}$$where${\displaystyle \mathbf{C}}$ is an arbitrary constant vector.

In Cartesian coordinates, the divergence of acontinuously differentiablevector field${\displaystyle \mathbf{F}=F_x\mathbf{i}+F_y\mathbf{j}+F_z\mathbf{k}}$ is the scalar-valued function:

As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.

The divergence of atensor field${\displaystyle \mathbf{T}}$ of non-zero orderk is written as${\displaystyle \mathrm{div}(\mathbf{T})=\mathrm{\nabla }\cdot \mathbf{T}}$, acontraction of a tensor field of orderk − 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 ofouter products and using the identity,$${\displaystyle \mathrm{\nabla }\cdot \left(\mathbf{A}\otimes \mathbf{T}\right)=\mathbf{T}(\mathrm{\nabla }\cdot \mathbf{A})+(\mathbf{A}\cdot \mathrm{\nabla })\mathbf{T}}$$where${\displaystyle \mathbf{A}\cdot \mathrm{\nabla }}$ is thedirectional derivative in the direction of${\displaystyle \mathbf{A}}$ multiplied by its magnitude. Specifically, for the outer product of two vectors,^[2]$${\displaystyle \mathrm{\nabla }\cdot \left(\mathbf{A}{\mathbf{B}}^{\mathsf{\text{T}}}\right)=\mathbf{B}(\mathrm{\nabla }\cdot \mathbf{A})+(\mathbf{A}\cdot \mathrm{\nabla })\mathbf{B}.}$$

For a tensor field${\displaystyle \mathbf{T}}$ of orderk > 1, the tensor field${\displaystyle \mathrm{\nabla }\cdot \mathbf{T}}$ of orderk − 1 is defined by the recursive relation$${\displaystyle (\mathrm{\nabla }\cdot \mathbf{T})\cdot \mathbf{C}=\mathrm{\nabla }\cdot (\mathbf{T}\cdot \mathbf{C})}$$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:wherei,j, andk are theunit vectors for thex-,y-, andz-axes, respectively.

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

InEinstein notation, the vector field${\displaystyle \mathbf{F}=\left(\begin{array}{c}{F}_1,F_2,F_3\end{array}\right)}$ has curl given by:$${\displaystyle \mathrm{\nabla }\times \mathbf{F}={\epsilon }^{ijk}{\mathbf{e}}_i\frac{\mathrm{\partial }{F}_k}{\mathrm{\partial }{x}_j}}$$where${\displaystyle \epsilon }$ = ±1 or 0 is theLevi-Civita parity symbol.

For a tensor field${\displaystyle \mathbf{T}}$ of orderk > 1, the tensor field${\displaystyle \mathrm{\nabla }\times \mathbf{T}}$ of orderk is defined by the recursive relation$${\displaystyle (\mathrm{\nabla }\times \mathbf{T})\cdot \mathbf{C}=\mathrm{\nabla }\times (\mathbf{T}\cdot \mathbf{C})}$$where${\displaystyle \mathbf{C}}$ is an arbitrary constant vector.

A tensor field of order greater than one may be decomposed into a sum ofouter products, and then the following identity may be used:$${\displaystyle \mathrm{\nabla }\times \left(\mathbf{A}\otimes \mathbf{T}\right)=(\mathrm{\nabla }\times \mathbf{A})\otimes \mathbf{T}-\mathbf{A}\times (\mathrm{\nabla }\mathbf{T}).}$$Specifically, for the outer product of two vectors,^[3]$${\displaystyle \mathrm{\nabla }\times \left(\mathbf{A}{\mathbf{B}}^{\mathsf{\text{T}}}\right)=(\mathrm{\nabla }\times \mathbf{A}){\mathbf{B}}^{\mathsf{\text{T}}}-\mathbf{A}\times (\mathrm{\nabla }\mathbf{B}).}$$

InCartesian coordinates, the Laplacian of a function${\displaystyle f(x,y,z)}$ is$${\displaystyle \mathrm{\Delta }f={\mathrm{\nabla }}^{2}f=(\mathrm{\nabla }\cdot \mathrm{\nabla })f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}.}$$

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 aharmonic function. That is,$${\displaystyle \mathrm{\Delta }f=0.}$$

For atensor field,${\displaystyle \mathbf{T}}$, the Laplacian is generally written as:$${\displaystyle \mathrm{\Delta }\mathbf{T}={\mathrm{\nabla }}^2\mathbf{T}=(\mathrm{\nabla }\cdot \mathrm{\nabla })\mathbf{T}}$$and is a tensor field of the same order.

For a tensor field${\displaystyle \mathbf{T}}$ of orderk > 0, the tensor field${\displaystyle {\mathrm{\nabla }}^2\mathbf{T}}$ of orderk is defined by the recursive relation$${\displaystyle \left({\mathrm{\nabla }}^2\mathbf{T}\right)\cdot \mathbf{C}={\mathrm{\nabla }}^2(\mathbf{T}\cdot \mathbf{C})}$$where${\displaystyle \mathbf{C}}$ is an arbitrary constant vector.

InFeynman subscript notation,$${\displaystyle {\mathrm{\nabla }}_{\mathbf{B}}\left(\mathbf{A}\cdot \mathbf{B}\right)=\mathbf{A}\times \left(\mathrm{\nabla }\times \mathbf{B}\right)+\left(\mathbf{A}\cdot \mathrm{\nabla }\right)\mathbf{B}}$$where the notation ∇_B means the subscripted gradient operates on only the factorB.^[4][5][6]

More general but similar is theHestenesoverdot notation ingeometric algebra.^[7][8] The above identity is then expressed as:$${\displaystyle \dot{\mathrm{\nabla }}\left(\mathbf{A}\cdot \dot{\mathbf{B}}\right)=\mathbf{A}\times \left(\mathrm{\nabla }\times \mathbf{B}\right)+\left(\mathbf{A}\cdot \mathrm{\nabla }\right)\mathbf{B}}$$where overdots define the scope of the vector derivative. The dotted vector, in this caseB, is differentiated, while the (undotted)A is held constant.

The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identityC⋅(A×B) = (C×A)⋅B:

An alternative method is to use the Cartesian components of the del operator as follows (withimplicit summation over the index i):

Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term (i.e., the operators must be nested). The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule.For example, from the identityA⋅(B×C) = (A×B)⋅Cwe may deriveA⋅(∇×C) = (A×∇)⋅C but not ∇⋅(B×C) = (∇×B)⋅C,nor fromA⋅(B×A) = 0 may we deriveA⋅(∇×A) = 0.On the other hand, a subscripted del operates on all occurrences of the subscript in the term, so thatA⋅(∇_A×A) = ∇_A⋅(A×A) = ∇⋅(A×A) = 0.Also, fromA×(A×C) =A(A⋅C) − (A⋅A)C we may derive ∇×(∇×C) = ∇(∇⋅C) − ∇²C,but from (Aψ)⋅(Aφ) = (A⋅A)(ψφ) we may not derive (∇ψ)⋅(∇φ) = ∇²(ψφ).

A subscriptc on a quantity indicates that it is temporarily considered to be a constant. Since a constant is not a variable, when the substitution rule (see the preceding paragraph) is used it, unlike a variable, may be moved into or out of the scope of a del operator, as in the following example:^[9]

Another way to indicate that a quantity is a constant is to affix it as a subscript to the scope of a del operator, as follows:^[10]$${\displaystyle \mathrm{\nabla }{\left(\mathbf{A}\cdot \mathbf{B}\right)}_{\mathbf{A}}=\mathbf{A}\times \left(\mathrm{\nabla }\times \mathbf{B}\right)+\left(\mathbf{A}\cdot \mathrm{\nabla }\right)\mathbf{B}}$$

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

For scalar fields${\displaystyle \psi }$,${\displaystyle \varphi }$ and vector fields${\displaystyle \mathbf{A}}$,${\displaystyle \mathbf{B}}$, we have the following derivative identities.

We have the following generalizations of theproduct rule in single-variablecalculus.

Let${\displaystyle f(x)}$ be a one-variable function from scalars to scalars,${\displaystyle \mathbf{r}(t)=(x_1(t),\dots ,x_n(t))}$ aparametrized curve,${\displaystyle \varphi :{\mathbb{R}}^n\to \mathbb{R}}$ 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-variablechain rule.

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

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

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

where${\displaystyle {\mathbf{J}}_{\mathbf{A}}=(\mathrm{\nabla }\mathbf{A}{)}^{\mathsf{\text{T}}}=(\mathrm{\partial }{A}_i/\mathrm{\partial }{x}_j{)}_{ij}}$ denotes theJacobian matrix of the vector field${\displaystyle \mathbf{A}=(A_1,\dots ,A_n)}$.

Alternatively, using Feynman subscript notation,

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

See these notes.^[11]

As a special case, whenA =B,

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

The generalization of thedot product formula toRiemannian manifolds is a defining property of aRiemannian connection, which differentiates a vector field to give a vector-valued1-form.

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

Thedivergence of the curl ofany continuously twice-differentiablevector fieldA is always zero:$${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0}$$

This is a special case of the vanishing of the square of theexterior derivative in theDe Rhamchain complex.

TheLaplacian of a scalar field is the divergence of its gradient:$${\displaystyle \mathrm{\Delta }\psi ={\mathrm{\nabla }}^2\psi =\mathrm{\nabla }\cdot (\mathrm{\nabla }\psi )}$$The result is a scalar quantity.

The divergence of a vector fieldA is a scalar, and the divergence of a scalar quantity is undefined. Therefore,$${\displaystyle \mathrm{\nabla }\cdot (\mathrm{\nabla }\cdot \mathbf{A})\text{ is undefined.}}$$

Thecurl of thegradient ofany continuously twice-differentiablescalar field${\displaystyle \phi }$ (i.e.,differentiability class${\displaystyle C^2}$) is always thezero vector:$${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\phi )=\mathbf{0}.}$$

It can be easily proved by expressing${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\phi )}$ in aCartesian coordinate system withSchwarz'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 theexterior derivative in theDe Rhamchain complex.

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

Here ∇² is thevector Laplacian operating on the vector fieldA.

Thedivergence of a vector fieldA is a scalar, and the curl of a scalar quantity is undefined. Therefore,$${\displaystyle \mathrm{\nabla }\times (\mathrm{\nabla }\cdot \mathbf{A})\text{ is undefined.}}$$

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.

Below, thecurly 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-dimensionalboundaryS = ∂V (aclosed surface):

• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \psi d\mathbf{S}={\iiint }_V\mathrm{\nabla }\psi dV}$
• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \mathbf{A}\cdot d\mathbf{S}={\iiint }_V\mathrm{\nabla }\cdot \mathbf{A}dV}$ (divergence theorem)
• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \mathbf{A}\times d\mathbf{S}=-{\iiint }_V\mathrm{\nabla }\times \mathbf{A}dV}$
• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \psi \mathrm{\nabla }\phi \cdot d\mathbf{S}={\iiint }_V\left(\psi {\mathrm{\nabla }}^2\phi +\mathrm{\nabla }\phi \cdot \mathrm{\nabla }\psi \right)dV}$ (Green's first identity)
• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \left(\psi \mathrm{\nabla }\phi -\phi \mathrm{\nabla }\psi \right)\cdot d\mathbf{S}=}$${\displaystyle \mathrm{\partial }V}$${\displaystyle \left(\psi \frac{\mathrm{\partial }\phi }{\mathrm{\partial }n}-\phi \frac{\mathrm{\partial }\psi }{\mathrm{\partial }n}\right)dS}$${\displaystyle {\displaystyle ={\iiint }_V\left(\psi {\mathrm{\nabla }}^2\phi -\phi {\mathrm{\nabla }}^2\psi \right)dV}}$ (Green's second identity)
• ${\displaystyle {\iiint }_V\mathbf{A}\cdot \mathrm{\nabla }\psi dV=}$${\displaystyle \mathrm{\partial }V}$${\displaystyle \psi \mathbf{A}\cdot d\mathbf{S}-{\iiint }_V\psi \mathrm{\nabla }\cdot \mathbf{A}dV}$ (integration by parts)
• ${\displaystyle {\iiint }_V\psi \mathrm{\nabla }\cdot \mathbf{A}dV=}$${\displaystyle \mathrm{\partial }V}$${\displaystyle \psi \mathbf{A}\cdot d\mathbf{S}-{\iiint }_V\mathbf{A}\cdot \mathrm{\nabla }\psi dV}$ (integration by parts)
• ${\displaystyle {\iiint }_V\mathbf{A}\cdot \left(\mathrm{\nabla }\times \mathbf{B}\right)dV=-}$${\displaystyle \mathrm{\partial }V}$${\displaystyle \left(\mathbf{A}\times \mathbf{B}\right)\cdot d\mathbf{S}+{\iiint }_V\left(\mathrm{\nabla }\times \mathbf{A}\right)\cdot \mathbf{B}dV}$ (integration by parts)
• ${\displaystyle \mathrm{\partial }V}$${\displaystyle \mathbf{A}\times \left(d\mathbf{S}\cdot \left(\mathbf{B}{\mathbf{C}}^{\mathsf{\text{T}}}\right)\right)={\iiint }_V\mathbf{A}\times \left(\mathrm{\nabla }\cdot \left(\mathbf{B}{\mathbf{C}}^{\mathsf{\text{T}}}\right)\right)dV+{\iiint }_V\mathbf{B}\cdot (\mathrm{\nabla }\mathbf{A})\times \mathbf{C}dV}$^[14]
• ${\displaystyle {\iiint }_V\left(\mathrm{\nabla }\cdot \mathbf{B}+\mathbf{B}\cdot \mathrm{\nabla }\right)\mathbf{A}dV=}$${\displaystyle \mathrm{\partial }V}$${\displaystyle \left(\mathbf{B}\cdot d\mathbf{S}\right)\mathbf{A}}$^[15]

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

• ${\displaystyle {\oint }_{\mathrm{\partial }S}\mathbf{A}\cdot d\mathit{\ell }={\iint }_S\left(\mathrm{\nabla }\times \mathbf{A}\right)\cdot d\mathbf{S}}$ (Stokes' theorem)
• ${\displaystyle {\oint }_{\mathrm{\partial }S}\psi d\mathit{\ell }=-{\iint }_S\mathrm{\nabla }\psi \times d\mathbf{S}}$
• ${\displaystyle {\oint }_{\mathrm{\partial }S}\mathbf{A}\times d\mathit{\ell }=-{\iint }_S\left(\mathrm{\nabla }\mathbf{A}-(\mathrm{\nabla }\cdot \mathbf{A})\mathbf{1}\right)\cdot d\mathbf{S}=-{\iint }_S\left(d\mathbf{S}\times \mathrm{\nabla }\right)\times \mathbf{A}}$
• ${\displaystyle {\oint }_{\mathrm{\partial }S}\mathbf{A}\times (\mathbf{B}\times d\mathit{\ell })={\iint }_S\left(\mathrm{\nabla }\times \left(\mathbf{A}{\mathbf{B}}^{\mathsf{\text{T}}}\right)\right)\cdot d\mathbf{S}+{\iint }_S\left(\mathrm{\nabla }\cdot \left(\mathbf{B}{\mathbf{A}}^{\mathsf{\text{T}}}\right)\right)\times d\mathbf{S}}$^[16]
• ${\displaystyle {\oint }_{\mathrm{\partial }S}(\mathbf{B}\cdot d\mathit{\ell })\mathbf{A}={\iint }_S(d\mathbf{S}\cdot \left[\mathrm{\nabla }\times \mathbf{B}-\mathbf{B}\times \mathrm{\nabla }\right])\mathbf{A}}$^[17]

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

${\displaystyle \mathrm{\partial }S}$${\displaystyle \mathbf{A}\cdot d\mathit{\ell }=-}$${\displaystyle \mathrm{\partial }S}$${\displaystyle \mathbf{A}\cdot d\mathit{\ell }.}$

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

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