Calculus/Vector calculus identities

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Vector calculus identities

In this chapter, numerous identities related to the gradient (), directional derivative (, ), divergence (), Laplacian (, ), and curl () will be derived.

Notation[edit]

To simplify the derivation of various vector identities, the following notation will be utilized:

  • The coordinates will instead be denoted with respectively.
  • Given an arbitrary vector , then will denote the entry of where . All vectors will be assumed to be denoted by Cartesian basis vectors () unless otherwise specified: .
  • Given an arbitrary expression that assigns a real number to each index , then will denote the vector whose entries are determined by . For example, .
  • Given an arbitrary expression that assigns a real number to each index , then will denote the sum . For example, .
  • Given an index variable , will rotate forwards by 1, and will rotate forwards by 2. In essence, and . For example, .

As an example of using the above notation, consider the problem of expanding the triple cross product .

Therefore:

As another example of using the above notation, consider the scalar triple product

The index in the above summations can be shifted by fixed amounts without changing the sum. For example, . This allows:

which establishes the cyclical property of the scalar triple product.

Gradient Identities[edit]

Given scalar fields, and , then .

Derivation


Given scalar fields and , then . If is a constant , then .

Derivation


Given vector fields and , then

Derivation


Given scalar fields and an input function , then .

Derivation


Directional Derivative Identities[edit]

Given vector fields and , and scalar field , then .

When is a vector field, it is also the case that: .

Derivation

For scalar fields:

For vector fields:


Given vector field , and scalar fields and , then .

When is a vector field, it is also the case that: .

Derivation

For scalar fields:

For vector fields:


Given vector field , and scalar fields and , then .

When and are vector fields, it is also the case that: .

Derivation

For scalar fields:

For vector fields:


Given vector field , and scalar fields and , then

If is a vector field, it is also the case that:

Derivation

For scalar fields:

For vector fields:


Given vector fields , , and , then

Derivation


Given vector fields , , and , then

Derivation


Divergence Identities[edit]

Given vector fields and , then .

Derivation


Given a scalar field and a vector field , then . If is a constant , then . If is a constant , then .

Derivation


Given vector fields and , then .

Derivation

In the above derivation, the third equality is established by cycling the terms inside a sum. For example: by replacing with . Different terms can be cycled independently:


The following identity is a very important property regarding vector fields which are the curl of another vector field. A vector field which is the curl of another vector field is divergence free. Given vector field , then

Derivation


Laplacian Identities[edit]

Given scalar fields and , then

When and are vector fields, it is also the case that:

Derivation

For scalar fields:

For vector fields:


Given scalar fields and , then

When is a vector field, it is also the case that

Derivation

For scalar fields:

For vector fields:


Curl Identities[edit]

Given vector fields and , then

Derivation


Given scalar field and vector field , then . If is a constant , then . If is a constant , then .

Derivation


Given vector fields and , then

Derivation


The following identity is a very important property of vector fields which are the gradient of a scalar field. A vector field which is the gradient of a scalar field is always irrotational. Given scalar field , then

Derivation


The following identity is a complex, yet popular identity used for deriving the Helmholtz decomposition theorem. Given vector field , then

Derivation


Basis Vector Identities[edit]

The Cartesian basis vectors , , and are the same at all points in space. However, in other coordinate systems like cylindrical coordinates or spherical coordinates, the basis vectors can change with respect to position.

In cylindrical coordinates, the unit-length mutually perpendicular basis vectors are , , and at position which corresponds to Cartesian coordinates .

In spherical coordinates, the unit-length mutually perpendicular basis vectors are , , and at position which corresponds to Cartesian coordinates .

It should be noted that is the same in both cylindrical and spherical coordinates.

This section will compute the directional derivative and Laplacian for the following vectors since these quantities do not immediately follow from the formulas established for the directional derivative and Laplacian for scalar fields in various coordinate systems.

which is the unit length vector that points away from the z-axis and is perpendicular to the z-axis.
which is the unit length vector that points around the z-axis in a counterclockwise direction and is both parallel to the xy-plane and perpendicular to the position vector projected onto the xy-plane.
which is the unit length vector that points away from the origin.
which is the unit length vector that is perpendicular to the position vector and points "south" on the surface of a sphere that is centered on the origin.

The following quantities are also important:

which is the perpendicular distance from the z-axis.
which is the azimuth: the counterclockwise angle of the position vector relative to the x-axis after being projected onto the xy-plane.
which is the distance from the origin.
which is the angle of the position vector to the z-axis.

Vector [edit]

only changes with respect to : .

Given vector field where is always orthogonal to , then

Derivation

Using cylindrical coordinates, let

The cylindrical coordinate version of the directional derivative gives:


Derivation

Using the cylindrical coordinate version of the Laplacian,


Vector [edit]

only changes with respect to : .

Given vector field where is always orthogonal to , then

Derivation

Using cylindrical coordinates, let

The cylindrical coordinate version of the directional derivative gives:


Derivation

Using the cylindrical coordinate version of the Laplacian,


Vector [edit]

changes with respect to and : and

Given vector field , then

Derivation

The spherical coordinate version of the directional derivative gives:


Derivation

The spherical coordinate version of the Laplacian gives:


Vector [edit]

changes with respect to and : and

Given vector field , then

Derivation

The spherical coordinate version of the directional derivative gives:


Derivation

The spherical coordinate version of the Laplacian gives: