§ **2
****Field Theory Preliminary**

1.
Basic Concepts of Field Theory and Gradient, Divergence and Curl

[ scalar field ] Each point *M* ( *x* , *y* , *z* ) in the space area *D* corresponds to a quantity value ( *x* , *y* , *z* ) , which constitutes a scalar field on this space area *D , using the point **M* ( *x,y , z* ) is represented by the scalar function ( *x* , *y* , *z* ) . If the position of *M* is determined by the vector radius ** r** , the scalar can be regarded as a function of the variable vector

For example, the temperature field *u* ( *x* , *y* , *z* ), the density field , and the electric potential field *e* ( *x* , *y* , *z* ) are all scalar fields ._{}

[ Vector field ] Each point *M* ( *x* , *y* , *z* ) in space area *D* corresponds to a vector value ** r** (

** r** (

** **For example, the velocity field ( *x* , *y* , *z* ) , the electric field ** E** (

As in the case of scalar fields, the concept of a vector field is essentially the same as that of a vector function . These terms ( scalar field, vector field ) are used to preserve their own origin and physical meaning .

[ gradient ]

grad _{}= ( _{}, , ) = = ** i** +

In the formula = ** i** +

The direction of grad coincides with _{}the normal direction ** N** of the isometric plane =

Gradients have properties:

grad( _{}_{}+ ) = grad + grad ( , is a constant )_{}_{}_{}_{}_{} _{}_{}

grad( ) _{}_{}= grad + grad_{}_{}_{}_{}

grad *F* ( ) _{}=_{}

[ Directional Derivative ]

_{}= ** l** · grad

where ** l** = (cos , cos , cos ) is

The directional derivative is the law of change in the direction ** l** , which is equal to the projection of the gradient in the direction

[ divergence ]

div ** r** = + + =

where is the Hamiltonian ._{}

Divergence has the property:

div( _{}** a** +

div( _{}** a** ) = div

div *( *** a** ×

[ curl ]

rot ** r** = ( )

where is Hamiltonian operator, curl is also called vorticity, rot ** r** is denoted as curl

Curl has the properties:

rot( _{}** a** +

rot( _{}** a** ) = rot

rot( ** a** ×

[ Gradient, divergence, curl mixed operation ] The operation grad acts on a scalar field to generate a vector field grad , the operation div acts on a vector field ** r** to generate a scalar field div

rot
** r** . The mixed operation formula of these three operations is as follows:

div rot ** r** = 0

rot grad _{}= **0**

div grad _{}= + + =_{} _{}_{}_{}_{}

grad div ** r** = (

rot rot ** r** = × ( ×

div grad( + ) = div grad + div grad ( _{}_{}_{}_{}_{}_{}_{} _{}, is a constant )_{}

div grad( )= div grad _{}_{}_{}_{}+ div grad +2 grad · grad_{}_{}_{}_{}

grad div ** r** - rot rot

where is the Hamiltonian, and **=** = **2 **^{is} the Laplace operator . _{}_{}_{}_{}_{}^{}

[ Potential field ( conservation field )] If the vector field ** r** (

** r** =grad

Then ** r** is called the potential field, and the scalar function is called the potential function of

The necessary and sufficient conditions for the vector field ** r** to be a potential field are: rot

_{} = , = , =_{}_{}_{}_{}_{}

Potential function calculation formula

_{}( *x* , *y* , *z* ) = ( *x *_{0} , *y *_{0} , *z *_{0} ) + +_{}_{}_{}_{}_{}_{}

+_{}

[ No scatter ( tubular field )] If the divergence of the vector field ** r** is zero, that is, div

** T** =

where *r* is the distance from *dV* to *M* , and the integration is performed over the entire space .

[ Irrotational field ] If the curl of the vector field ** r** is zero, that is, rot

_{}=-_{} _{}

where *r* is the distance from d *V* to *M* , and the integration is performed over the entire space .

2.
Expressions of gradient, divergence and curl in different coordinate systems

1 . unit vector transformation

[ General formula ] Assume that *x* = *f* ( ), *y* = *g* ( ), *z* = *h* ( ) _{}_{}_{}to continuously map a region of ( ) _{}space to a region *D* of ( *x* , *y* , *z* ) space , and Assuming that *f* , *g* , *h* have continuous partial derivatives, because the correspondence is one-to-one, we have

_{}_{}= ( *x* , *y* , *z* ) ,_{}_{}

Assuming that there are also continuous partial derivatives, then we have_{}

_{}

or inverse transform

_{}

The unit vectors along the d *x* , *dy* , d z directions are denoted as ** i** ,

_{}

[ Unit vector of cylindrical coordinate system ] For cylindrical coordinate system ( Fig. 8.11)

_{} _{}

The unit vector is

_{}

Their partial derivatives are

_{}

[ Unit vector of spherical coordinate system ] For spherical coordinate system ( Fig. 8.12)

_{}
_{}

The unit vector is

_{}

Their partial derivatives are

_{}

** **2 . Coordinate transformation of vectors

[ General formula ] A vector expressed by the ( *x* , *y* , *z* ) coordinate system can be expressed by the ( ) coordinate system :_{}

_{}= ( _{}, * _{y}* ,

in the formula

_{}

[ Exchange of Cylindrical Coordinate System and Cartesian Coordinate System ] Transformation Formula from Cylindrical Coordinate System to Cartesian Coordinate System

_{}

Transformation formula from Cartesian coordinate system to cylindrical coordinate system

_{}

[ Exchange of spherical coordinate system and rectangular coordinate system ] Transformation formula from spherical coordinate system to rectangular coordinate system

_{}

Transformation formula from rectangular coordinate system to spherical coordinate system

_{}

3 . Expressions of various operators in different coordinate systems

Let *U* = *U* ( *x* , *y* , *z* ) be a scalar function and ** V** =

[ Expressions of various operators in the cylindrical coordinate system ]

Hamiltonian = + + _{}_{}_{}_{}

Gradient grad *U* = *U* = + + _{}_{}_{}_{}

Divergence div ** V** = ·

Curl rot ** V** = ×

Laplacian *U* = div grad *U* = _{}_{}

[ Expressions of various operators in spherical coordinates ]

Hamiltonian = + + _{}_{}_{}_{}

Gradient grad *U* = *U* = + + * *_{}_{}_{}_{}

Divergence div ** V** = ·

Curl rot ** V** = ×

+_{}_{}

+_{}_{}

Laplacian *U* = div grad *U* _{}

=_{}

3.
Curve integral, surface integral and volume derivative

[ Curve integral of a vector and its calculation formula ] The curvilinear integral of a vector field ** r** (

_{}** r** (

where *ri ** _{-1}* =

_{}From *A* to *B* ( Fig. 8.13)

If the vector function ** R** (

continuous function ), the curve is also continuous and has continuous rotation_{}

tangent , the curve integral

_{}

exist .

If ** R** (

The work done by a force R when ** a** particle moves along

The formula for calculating the vector curve integral is as follows:

_{}=_{}

_{}= _{}+ ( Figure 8.14)_{}

_{}= -_{}

_{}= _{}+_{}

_{}= *k* ( *k*_{} is a constant )

[ Circulation of the vector ] If *G* is a closed curve, then the curve integral along the curve *G *

_{}=_{}

**It is called the circulation of the vector field **** R** (

The circulation of the potential field along any closed curve is equal to zero . If ** R** (

_{}= _{}= ( *B* ) - ( *A* )_{}_{}

It has nothing to do with the path connecting points *A* and *B* , but only depends on the path between points *A* and *B.*

position ( Figure 8.15).

[ Surface integral of a vector ] Let *S* be a surface, let ** N** = denote the normal unit vector of a point on the surface

1 Flux ( or flow ) of a scalar field^{ }

_{}d ** S** = d

where *S _{yz}* ,

*Projection on the Oxy* plane . * _{}*The sign of

When looking to the side, what you see is the front of the surface *S* , consider *S _{xy}* to be positive, if

If you see the opposite side of the surface, consider *S _{xy}* to be negative ( Figure 8.16).

2 ^{ }Scalar Flux of a Vector Field

_{}** R** d

*The meanings of S _{yz}* etc. in the formula are the same as 1 .

3 ^{ }Vector flux of a vector field

_{}** R** ×d

*The meanings of S _{yz}* etc. in the formula are the same as 1 .

[ Volume Derivative of Vector ] If *S* is a closed surface enclosing volume *V* and contains point *r*** , then the ratio of** the surface integral ( d *S*** , **** R** d

1 The volume derivative of a scalar field is its gradient:^{} _{}

grad _{}=_{}

2 One of the volume derivatives of a ^{} vector field ** R** is its divergence:

div
** R** =

3 Another volume derivative of the ^{} vector field ** R** is its curl:

rot
** R** = -

4.
Integral Theorem of Vectors

[ Gaussian formula ]

_{}** R** d

which is

_{}

where *S* is the boundary surface of the space region *V , and *** N** = is

normal unit vector at a point on *S , *** R** (

There are continuous partial derivatives on *V* + *S.*

[ Stokes formula ]

_{}rot *R*** ·** d ** S** = rot

which is

_{}

=_{}

=_{}

In the formula, *S* is one side of a certain surface, *L* is the closed boundary curve of the surface *S* ( the positive direction of *L and *** N** form a right-handed system ). Each point of

[ Green formula ]

_{}· dS = *_*_{}

_{}· dS = *_*_{}

In the formula, *S* is the boundary surface of the space region *V* , which is two scalar functions. It has continuous partial derivatives on *S* and second-order continuous partial derivatives on *V.* It is the Laplace operator, especially_{}_{}

_{}· dS = *_*_{}

which is

_{}

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