This document discusses key concepts in vector calculus including paths, surfaces, volumes, and three important theorems - the gradient theorem, divergence theorem, and curl theorem. It defines paths, surfaces, and volumes and explains that they can be scalar or vector quantities depending on whether they represent lengths, areas, or volumes. It also introduces the concepts of differential length, area, and volume and expresses them mathematically in Cartesian, cylindrical and spherical coordinate systems. Finally, it states the mathematical formulations of the three theorems - the gradient theorem relates a line integral to function values, the divergence theorem relates a volume integral to a surface integral, and the curl theorem relates a surface integral to a closed line integral.

1. Series: EMF Theory
Lecture: # 0.13
Dr R S Rao
Professor, ECE
Paths, surfaces and volumes, gradient theorem, divergence theorem and curl theorem,
differential length, surface and volumes in different Coordinate systems
Passionate
Teaching
Joyful
Learning

2. Paths, Surfaces and Volumes
Electromagnetic
Fields
Vector
Calculus-III
2
• In integral calculus, volume, surface and paths are mathematical
entities with very clear definitions.
• The volume, by definition, is a region, always surrounded by a
closed surface.
• And surface, once again by definition is an entity always
surrounded by a closed path.
• The path, by definition, is a curve with direction. Usually, the terms
paths and curves are used to mean the same.

3. Paths, Surfaces and Volumes
Electromagnetic
Fields
Vector
Calculus-III
3
• Volumes/differential volumes are scalar quantities whereas
surfaces/differential surfaces and paths/differential lengths are
vectors.
• Direction of path is tangential to the path and indicated with an
arrow.
• For straight line path, direction remains same all along its length.
And for curved line paths, direction naturally varies from point to
point.
• Two types of paths are encountered in electromagnetics: open paths
and closed paths.
• In case of open paths, starting point and terminating points are
different whereas in case of closed paths they are same.

4. Electromagnetic
Fields
Vector
Calculus-III
4
• Direction of surface, by definition, is normal to the surface. If the
surface is a flat plane, the direction remains same at all points
over its surface.
• For curved surfaces, direction varies from point to point.
• At an arbitrary point over the surface, direction can be found by
wrapping the point with right hand fingers in the direction of the
enclosing curve, direction pointed by the thumb then giving
direction of the surface at that point.
Paths, Surfaces and Volumes

5. Electromagnetic
Fields
Vector
Calculus-III
5
Paths, Surfaces and Volumes

6. Electromagnetic
Fields
VECTOR
CALCULUS-II
6
Differential elements
•Differential length is vector joining two points whose
coordinates differ by differential amounts.
•Differential area is surface of a parallelepiped formed around
differential length as diagonal
•Differential volume is volume of the parallelepiped.

7. Electromagnetic
Fields
VECTOR
CALCULUS-II
7
Differential elements
Differential length:
ˆ ˆ ˆ
Differential areas:
ˆ
ˆ
ˆ
Differential volume:
x
y
z
d dx dy dz
d dydz
d dzdx
d dxdy
d dxdydz

  




l x y z
a x
a y
a z
Cartesian system :::

8. Electromagnetic
Fields
VECTOR
CALCULUS-II
8
Differential elements
Differential length:
ˆ
ˆ ˆ
Differential areas:
ˆ
ˆ
ˆ
Differential volume:
z
d d d dz
d d dz
d d dz
d d d
d d d dz


  
 

  
   
  




l ρ z
a ρ
a
a z
Cylindrical system :::



9. Electromagnetic
Fields
VECTOR
CALCULUS-II
9
Differential elements
2
2
Differential length:
ˆ ˆ
ˆ sin
Differential areas:
ˆ
sin
ˆ
sin
ˆ
Differential volume:
sin
r
d dr rd r d
d r d d
d r drd
d rdrd
d r drd d


  
  
 

   
  




l r θ
a r
a θ
a
Spherical system :::



10. Calculus theorems
Electromagnetic
Fields
Vector
Calculus-III
10
•Fundamental/Gradient theorem of calculus
•Divergence/Gauss theorem
•Curl/Stokes theorem

11.      
l
b
a
f d f b f a
   

Electromagnetic
Fields
Vector
Calculus-III
11
Statement:
'the integral of the tangential component of gradient of a scalar
function along a path from a to b is the difference of the function
values at b and a i.e. f(b) and f(a)'.
Mathematically,
•It connects a line integral with values of function at end points.
Gradient theorem

12. d d

   
 
F F a
Electromagnetic
Fields
Vector
Calculus-III
12
Statement:
'The integral of the normal component of a vector (flux) over a closed
surface s is equal to the integral of the divergence of the same vector
throughout the volume v enclosed by the surface s.'
Mathematically,
•It connects a volume integral with closed surface integral.
Divergence theorem

13.   d d
    
 
F a F l
Electromagnetic
Fields
Vector
Calculus-III
13
Statement:
'The integral of the tangential component of a vector (circulation) around a
closed path c is equal to the integral of the normal component of the curl of
the same vector through any surface s enclosed by the path c .'
Mathematically ,
•It connects a surface integral with closed line integral.
Curl theorem

14. ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
14