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

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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

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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

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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

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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

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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

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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

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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 Rho

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  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 Phi

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  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 r

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  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 Theta

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  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: