Second Derivatives
• The gradient, the divergence and the curl are the
only first derivatives we can make with  , by
applying 
twice we can construct five species
of second derivatives.
• The gradient is a vector, so we can take the
divergence and curl of it.
(1) Divergence of gradient : .(A)  2 A (Laplacian)
(2) Curl of gradient:   (A)  0
o The divergence is a scalar, so we can take its
gradient.

(3) Gradient of divergence. (.A)
o The curl is a vector, so we can take its divergence
and curl.

(4) Divergence of a Curl. .(  A)  0 

(5) Curl of curl.   (  A)  (. A)  2 A
Integral Calculus
o Line ( Path) Integrals
o Surface (Flux) Integrals
o Volume Integrals
Line ( Path) Integrals:
Let v is a vector function
dl is the infinitesimal displacement
vector
Integral carried out along a specified
path from point a to point b gives line or
path integral.
b
 v.dl
a
Line integral depends critically on the
particular path taken from a to b.
If path forms a closed loop i.e. a = b, the
integral can be written as:
v
.
dl

There is a special type of vector which does
not depend on path, but is determined
entirely by the end points.
The force which have this property called
as conservative force.
Surface (Flux) Integrals:
Let v is a vector function
da is an infinitesimal patch of area
(direction is  to the surface)
If v is flow of a fluid (mass per unit area per
unit time)

S
v.da
: total mass per unit
time passing through
the surface
If the surface is closed:
v
.
da

o For closed surface, direction of da is
outward
o For open surfaces it’s arbitrary.
oSurface
integral
depends
particular surface chosen.
on
the
o There is a special class of vector function
for which integral is independent of the
surface, and is determined entirely by the
boundary line.
Volume Integrals:
Let T is a scalar function. (may be a vector also)
d is an infinitesimal volume element
 Td
v
,in Cartesian coordinates: d=dx.dy.dz
If T is the density of a substance
 Td
v
: total mass
Note: If T is a vector function then unit
vectors can be taken out from the
integral
The fundamental theorem of calculus:
b
b
df
a dx dx  a F ( x)dx  f (b)  f (a)
Where df/dx=F(x), we can think f(x) is a
function whose derivative is F(x)
and df=(df/dx).dx is the infinitesimal change in
f when we go from x to x+dx
Integral of a derivative over an interval (a to b)
is given by the value of the function at the end
points (boundaries)
In vector calculus:
derivatives
three
species
oGradient
oDivergent
oCurl
Each has its own
fundamental theorem
of
The fundamental theorem of Gradient
o Suppose we have a scalar
function
of
three
variables: T(x, y, z)
o By moving a distance
dl1, the function T will
change by an amount: dT  (T ).dl1
o So total change in T in going
from a to b along the path
selected:
b
 (T ).dl
a
o Also total change in T by going
from a to b can be represented by : T  b   T  a 
Or we can write:
b
 (T ).dl  T (b)  T (a)
a
This is called fundamental theorem of
Gradient.
It says “ The line integral of a derivative
(gradient) is given by the value of the
function at the boundaries”.
b
 (T ).dl  T (b)  T (a)
a
Left hand side tells: integral is path
dependent
Right hand side tells: integral is path
independent (depend only on end points)
Generally line integrals depend on path taken.
Note: But Gradients have the special
property that their integrals are path
independent.
Ques: If we are choosing a closed path, then
change in the scalar function using Gradient
theorem will be:
 (T ).dl  0
The Fundamental Theorem for Divergences
known as Gauss’ Divergence Theorem

v
(.v)d 

S
v.da
o It’
general
interpretation
(not
complete): The integral of a derivative
over a region is equal to the value of the
function at the boundary.
oComplete description:
Derivative is the form of (divergence)
Region is (volume)
Boundary is the surface that bounds the
volume (boundary indicates integral not
just the difference of two points).
Note:
 The boundary of a line:
2 end point
The boundary of a volume: Closed surface
It’s Geometrical Interpretation:
If v: flow of a incompressible fluid
The flux of v:
 (.v)d
v
= total amount of fluid
passing out through the surface, per unit time
Applications:
Gauss’s Theorem can be applied to any vector field
which obeys an inverse-square law, such as
gravitational, electrostatic attraction, and even
examples in quantum physics such as probability
density.
The fundamental theorem for Curls
known as Stokes’ Theorem
 (  v).da  
S
P
v.dl
Line integral
o It’ general interpretation (not complete):
The integral of a derivative over a region is
equal to the value of the function at the
boundary.
o Note: on the right hand side circle over
integral indicates perimeter of the
surface is closed, not the surface itself
oComplete description:
Derivative is the form of (curl)
Region is (patch of surface)
Boundary is the perimeter of the patch (due to
perimeter here boundary indicates a closed
line integral).
Outcomes:
1) S (  v).da depends only on the boundary line,
not on the particular surface used
2)  (  v).da =0 for any closed surface, since for a
closed surface, boundary line shrinks down to
a point.