Page 187 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 16.4 Gradient, Divergency, and Rotor

The tensorial nature of partial derivatives established by theorem 1.1 is a very useful feature.
We can apply it to extend the scope of classical operations of vector analysis. Let’s consider the
gradient, grad = . Usually the gradient operator is applied to scalar fields, i.e. to functions
 = (P) or  = (x , x , x ) in coordinate form:
2
3
1

a     . ...(1)
q
q
 x q
Note that in (1) we used a lower index q for a . This means that a = grad  is a covector. Indeed,
q
according to theorem 1.1, the nabla operator applied to a scalar field, which is tensor field of
type (0, 0), produces a tensor field of type (0, 1). In order to get the vector form of the gradient one
should raise index q:
3 3
q
i 
qi
qi
a   g a  g   . ...(2)
i
i 1 i 1


Let’s write (2) in the form of a differential operator (without applying to ):
3
qi
q
   g  i . ...(3)
i 1

In this form the gradient operator (3) can be applied not only to scalar fields, but also to vector
fields, covector fields and to any other tensor fields.
Usually in physics we do not distinguish between the vectorial gradient  and the covectorial
q
gradient  because we use orthonormal coordinates with ONB as a basis. In this case, dual
q
metric tensor is given by unit matrix (g =  ) and components of  and  coincide by value.
q
ij
ij
q
Divergency is the second differential operation of vector analysis. Usually it is applied to a
vector field and is given by formula:
3
i
div X    i X . ...(4)
i 1

As we see, (4) is produced by contraction (see section 16) from tensor qX . Therefore we can
i
generalize formula (4) and apply divergency operator to arbitrary tensor field with at least one
upper index:
3
(div X) ...............   s X ........s........ . ...(5)
............... 
..................
s 1

The Laplace operator is defined as the divergency applied to a vectorial gradient of something,
it is denoted by the triangle sign:  = div grad. From (3) and (5) for Laplace operator  we derive
the following formula:
3 3
ij
   g   j . ...(6)
i
i 1 j 1


Denote by the following differential operator:
1  2
= 2 2  .  ...(7)
c  t

180 LOVELY PROFESSIONAL UNIVERSITY