NationMaster - Encyclopedia: Vector calculus identities

The following identities are important in vector calculus: Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

Single operators (summary)

This section explicitly lists what some symbols mean for clarity.

Divergence

In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

Divergence of a vector field

For a vector field $mathbf{v}$ , divergence is generally written as

$operatorname{div}(mathbf{v}) = nabla cdot mathbf{v}$

and is a scalar field.

Divergence of a tensor

For a tensor $mathbf{A}$ , divergence is generally written as

$operatorname{div}(mathbf{A}) = nabla cdot mathbf{A}$

and is a vector.

Curl

For a vector field $mathbf{v}$ , curl is generally written as For other uses, see Curl (disambiguation). ...

$operatorname{curl}(mathbf{v}) = nabla times mathbf{v}$

and is a vector field.

Gradient

For other uses, see Gradient (disambiguation). ...

Gradient of a vector field

For a vector field $mathbf{v}$ , gradient is generally written as

$operatorname{grad}(mathbf{v}) = nabla mathbf{v}$

and is a tensor

Gradient of a scalar field

For a scalar field, $ψ$ , the gradient is generally written as

$operatorname{grad}(psi) = nabla psi$

and is a vector field.

Combinations of multiple operators

Curl of the gradient

The curl of the gradient of any scalar field $psi$ is always zero: For other uses, see Curl (disambiguation). ... For other uses, see Gradient (disambiguation). ... In mathematics and physics, a scalar field associates a scalar to every point in space. ...

$nabla times ( nabla psi ) = 0$

Divergence of the curl

The divergence of the curl of any vector field $mathbf{A}$ is always zero: In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

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

Divergence of the gradient

The Laplacian of a scalar field is defined as the divergence of the gradient: In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

$nabla cdot (nabla psi) = nabla^2 psi$

Note that the result is a scalar quantity.

Curl of the curl

$nabla times nabla times mathbf{A} = nabla(nabla cdot mathbf{A}) - nabla^{2}mathbf{A}$

Properties

Distributive property

$nabla cdot ( mathbf{A} + mathbf{B} ) = nabla cdot mathbf{A} + nabla cdot mathbf{B}$

$nabla times ( mathbf{A} + mathbf{B} ) = nabla times mathbf{A} + nabla times mathbf{B}$

Vector dot product

$nabla(mathbf{A} cdot mathbf{B}) = (mathbf{A} cdot nabla)mathbf{B} + (mathbf{B} cdot nabla)mathbf{A} + mathbf{A} times (nabla times mathbf{B}) + mathbf{B} times (nabla times mathbf{A})$

Vector cross product

$nabla cdot (mathbf{A} times mathbf{B}) = mathbf{B} cdot nabla times mathbf{A} - mathbf{A} cdot nabla times mathbf{B}$

$nabla times (mathbf{A} times mathbf{B}) = mathbf{A} (nabla cdot mathbf{B}) - mathbf{B} (nabla cdot mathbf{A}) + (mathbf{B} cdot nabla) mathbf{A} - (mathbf{A} cdot nabla) mathbf{B}$

Product of a scalar and a vector

$nabla cdot (psimathbf{A}) = mathbf{A} cdotnablapsi + psinabla cdot mathbf{A}$

$nabla times (psimathbf{A}) = psinabla times mathbf{A} - mathbf{A} times nablapsi$

Product Rule for the Gradient

The gradient of the product of two scalar fields $ψ$ and $φ$ follows the same form as the Product rule in single variable Calculus. In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... For other uses, see Calculus (disambiguation). ...

$nabla (psi , phi) = phi ,nabla psi + psi ,nabla phi$

More identities

$frac{1}{2} nabla A^2 = mathbf{A} times (nabla times mathbf{A}) + (mathbf{A} cdot nabla) mathbf{A}$

References

Constantine A. Balanis. Advanced Engineering Electromagnetics.

H. M. Schey (1997). Div Grad Curl and all that: An informal text on vector calculus. W. W. Norton & Company. ISBN 0-393-96997-5.

David J. Griffith (1999). Introduction to Electromagnetics. Prentice Hall. ISBN 0-13-805326-X.

Vector algebra (1100 words)
	A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
	The formal definition of a vector space is entirely abstract, like the concept of a field itself, and analogous to the concept of a module over a ring, of which it is a specialization.
	Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right.

Vector (spatial) - definition of Vector (spatial) in Encyclopedia (1910 words)
	A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry).
	Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations).
	The length or magnitude or norm of the vector a is denoted by

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