In vector calculus Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an, del is a vector In elementary mathematics, physics, and engineering, a Euclidean vector is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another, usually represented by the nabla symbol ∇. When applied to a function The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain defined on a one-dimensional domain, it denotes its standard derivative In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a vehicle with respect to time is the vehicle's instantaneous velocity as defined in calculus Calculus is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in. When applied to a field (a function defined on a multi-dimensional domain), del may denote the gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change (slope) of a scalar field In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space, the divergence In vector calculus, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated of a vector field In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space, or the curl In vector calculus, the curl is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point (rotational) of a vector field, depending on the way it is applied.

Strictly speaking, del is not a specific operator, but rather a convenient mathematical notation Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as numbers 1 and 2, function symbols sin and +; conceptual symbols, such for those three operators, that makes many equations An equation is a mathematical statement that asserts the equality of two expressions. Equations consist of the expressions that have to be equal on opposite sides of an equal sign, as in easier to write and remember. The del symbol can be interpreted as a vector of partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant . Partial derivatives are used in vector calculus and differential geometry operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the scalar product In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the dot that is often used to designate this operation; the alternative name scalar product emphasizes the scalar, dot product In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar, and cross product In mathematics, the cross product, vector product or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It has a vector result, a vector which is always perpendicular to both of the vectors being multiplied and the plane containing them. It has many applications in mathematics, engineering and physics, respectively, of the del "operator" with the field. These formal products may not commute In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the with other operators or products.

Definition

In the three-dimensional Cartesian coordinate system A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length R³ with coordinates (x, y, z), del is defined in terms of partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant . Partial derivatives are used in vector calculus and differential geometry operators as

where are the unit vectors in the respective coordinate directions.

Though this page chiefly treats del in three dimensions, this definition can be generalized to the n-dimensional Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity Rⁿ. In the Cartesian coordinate system A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length with coordinates (x₁, x₂, ..., x_n), del is:

where is the standard basis In mathematics, the standard basis for a Euclidean space consists of one unit vector pointing in the direction of each axis of the Cartesian coordinate system. For example, the standard basis for the Euclidean plane are the vectors in this space.

More compactly, using the Einstein summation notation In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulas. It was introduced by Albert Einstein in 1916, del is written as

Del can also be expressed in other coordinate systems, see for example del in cylindrical and spherical coordinates.

Notational uses

Del is used as a shorthand form to simplify many long mathematical expressions. It is most commonly used to simplify expressions for the gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change, divergence In vector calculus, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated, curl In vector calculus, the curl is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point, directional derivative, and Laplacian In mathematics the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. The Laplacian of a function ƒ, denoted by the symbols Δƒ, ∇2ƒ or ∇·∇ƒ, is the divergence of the gradient of ƒ, and as such represents the.

Gradient

The vector derivative of a scalar field In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space f is called the gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change, and it can be represented as:

It always points in the direction of greatest increase of f, and it has a magnitude The magnitude of a mathematical object is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs 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 h(x,y), the 2d projection of the gradient at a given location will be a vector in the xy-plane (sort of like 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 In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and adding up those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar do not turn out to be simple, as illustrated by:

Divergence

The divergence In vector calculus, divergence is an operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated of a vector field In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space is a scalar In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same point in space 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 a measure of that field's tendency to converge on or repel 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:

Curl

The curl In vector calculus, the curl is a vector operator that describes the rotation of a vector field. At every point in the field, the curl is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point of a vector field is a vector In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a subset of Euclidean space function that can be represented as:

The curl at a point is proportional to the on-axis torque to which a tiny pinwheel would be subjected if it were centered at that point.

The vector product operation can be visualised 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:

Directional derivative

The directional derivative of a scalar field f(x,y,z) in the direction is defined as:

This gives the change of a field f in the direction of a. In operator notation, the element in parentheses can be considered a single coherent unit; fluid dynamics In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, uses this convention extensively, terming it the convective derivative—the "moving" derivative of the fluid.

Laplacian

The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; it is defined as:

The Laplacian is ubiquitous throughout modern mathematical physics Mathematical physics is the scientific discipline concerned with the interface of mathematics and physics. The Journal of Mathematical Physics defines it as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.", appearing in Laplace's equation In mathematics, Laplace's equation is a partial differential equation named after Pierre-Simon Laplace who first studied its properties. The solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of, Poisson's equation In mathematics, Poisson's equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. It is named after the French mathematician, geometer and physicist Siméon-Denis Poisson. The Poisson equation is, the heat equation The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time. For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is, the wave equation The wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert,, and the Schrödinger equation In physics, specifically quantum mechanics, the Schrödinger equation is an equation that describes how the quantum state of a physical system changes in time. It is as central to quantum mechanics as Newton's laws are to classical mechanics—to name a few.

Tensor derivative

Del can also be applied to a vector field with the result being a tensor Tensors are geometric entities introduced into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices to higher orders. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute. The tensor derivative of a vector field is a 9-term second-rank tensor, but can be denoted simply as , where represents the dyadic product. This quantity is equivalent to the Jacobian matrix In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1, ..., ym(x1,...,xn). The partial derivatives of all these functions (if they of the vector field with respect to space.

For a small displacement , the change in the vector field is given by:

Second derivatives

When del operates on a scalar or vector, generally 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 In mathematics the Laplace operator or Laplacian, named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. The Laplacian of a function ƒ, denoted by the symbols Δƒ, ∇2ƒ or ∇·∇ƒ, is the divergence of the gradient of ƒ, and as such represents the 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 Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste. To ensure that an object is "well-behaved" mathematicians introduce, 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:

Precautions

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 del is replaced by any other vector. This is part of the tremendous value gained in representing this operator as a vector in its own right.

Though you can often replace del with a vector and obtain a vector identity, making those identities intuitive, the reverse is not necessarily reliable, because del does not often commute.

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 precise numerical magnitude and direction, del does not have a precise value for either until it is allowed to operate on something.

For that reason, identities involving del must be derived from scratch, not derived from pre-existing vector identities.

See also

• Table of mathematical symbols This is a listing of common symbols found within all branches of mathematics. Each symbol is listed in both HTML, which depends on appropriate fonts to be installed, and in TeX, as an image
• Navier-Stokes equations The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term , plus a pressure term
• Maxwell's equations
• Del in cylindrical and spherical coordinates
• Vector calculus identities

References

• Div, Grad, Curl, and All That, H. M. Schey, ISBN 0-393-96997-5
• Jeff Miller, Earliest Uses of Symbols of Calculus
• Cleve Moler, ed., "History of Nabla", NA Digest 98 (Jan. 26, 1998).

External links

• A survey of the improper use of ∇ in vector analysis (1994) Tai, Chen

Categories: Vector calculus | Mathematical notation | Differential operators

See Also:

What Links Here:

Recently Viewed Pages:

• Home
• History of Subcultures
• Geek Squad: Answers
• Blue Screen of Death: Answers
• Subculture
• Subcultures: News
• About Us
• Geek: Blogs
• BoardGameGeek
• Anorak (Slang)

Most Popular Pages:

• Fashion: News
• Perx Injector: Blogs
• Perx Combat Arms Hacks: Blogs
• Subcultures: Answers
• Geek Squad: Answers
• Home
• Perx: Blogs
• Blue Screen of Death: Answers
• 0x0000007E 0xc0000005: Answers
• Geek: Answers

Related 'Del' Searches: