Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operators depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field.
Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation for those three operators that makes many equations easier to write and remember. The del symbol (or nabla) can be interpreted as a vector of partial derivative operators; and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product, and a cross product, respectively, of the "del operator" with the field. These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
In the Cartesian coordinate system Rn with coordinates and standard basis , del is defined in terms of partial derivative operators as
In three-dimensional Cartesian coordinate system R3 with coordinates and standard basis or unit vectors of axes , del is written as
Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.
Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient, divergence, curl, directional derivative, and Laplacian.
The vector derivative of a scalar field is called the gradient, and it can be represented as:
It always points in the direction of greatest increase of , and it has a magnitude equal to the maximum rate of increase at the point—just like a standard derivative. In particular, if a hill is defined as a height function over a plane , the gradient at a given location will be a vector in the xy-plane (visualizable as an arrow on a map) pointing along the steepest direction. The magnitude of the gradient is the value of this steepest slope.
In particular, this notation is powerful because the gradient product rule looks very similar to the 1d-derivative case:
However, the rules for dot products do not turn out to be simple, as illustrated by:
The divergence of a vector field is a scalar function that can be represented as:
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it is a measure of that field's tendency to converge toward or diverge from a point.
The power of the del notation is shown by the following product rule:
The formula for the vector product is slightly less intuitive, because this product is not commutative:
The curl of a vector field is a vector function that can be represented as:
The curl at a point is proportional to the on-axis torque that a tiny pinwheel would be subjected to if it were centred at that point.
The vector product operation can be visualized as a pseudo-determinant:
Again the power of the notation is shown by the product rule:
Unfortunately the rule for the vector product does not turn out to be simple:
The directional derivative of a scalar field in the direction is defined as:
This gives the rate of change of a field in the direction of . In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics uses this convention extensively, terming it the convective derivative —the "moving" derivative of the fluid.
Note that is an operator that takes scalar to a scalar. It can be extended to operate on a vector, by separately operating on each of its components.
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian coordinate systems it is defined as:
and the definition for more general coordinate systems is given in vector Laplacian.
The Laplacian is ubiquitous throughout modern mathematical physics, appearing for example in Laplace's equation, Poisson's equation, the heat equation, the wave equation, and the Schrödinger equation.
While usually represents the Laplacian, sometimes also represents the Hessian matrix. The former refers to the inner product of , while the latter refers to the dyadic product of :
So whether refers to a Laplacian or a Hessian matrix depends on the context.
Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as , where represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.
For a small displacement , the change in the vector field is given by:
For vector calculus:
For matrix calculus (for which can be written ):
Another relation of interest (see e.g. Euler equations ) is the following, where is the outer product tensor:
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field f or a vector field v; the use of the scalar Laplacian and vector Laplacian gives two more:
These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved [ clarification needed ], two of them are always zero:
Two of them are always equal:
The 3 remaining vector derivatives are related by the equation:
And one of them can even be expressed with the tensor product, if the functions are well-behaved:
Most of the above vector properties (except for those that rely explicitly on del's differential properties—for example, the product rule) rely only on symbol rearrangement, and must necessarily hold if the del symbol is replaced by any other vector. This is part of the value to be gained in notationally representing this operator as a vector.
Though one can often replace del with a vector and obtain a vector identity, making those identities mnemonic, the reverse is not necessarily reliable, because del does not commute in general.
A counterexample that relies on del's failure to commute:
A counterexample that relies on del's differential properties:
Central to these distinctions is the fact that del is not simply a vector; it is a vector operator. Whereas a vector is an object with both a magnitude and direction, del has neither a magnitude nor a direction until it operates on a function.
For that reason, identities involving del must be derived with care, using both vector identities and differentiation identities such as the product rule.
In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a field is formally defined as the circulation density at each point of the field.
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field whose value at a point is the vector whose components are the partial derivatives of at . That is, for , its gradient is defined at the point in n-dimensional space as the vector:
An exact sequence is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.
The vorticity equation of fluid dynamics describes evolution of the vorticity ω of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid . The equation is:
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p).
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. It allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus.
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
A vector operator is a differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl:
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
Scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.
The following are important identities involving derivatives and integrals in vector calculus.
In differential calculus, there is no single uniform notation for differentiation. Instead, several different notations for the derivative of a function or variable have been proposed by different mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation are listed below.
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms.
In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.