2. INTRODUCTION
Vector calculus was developed by J.WILLARD GIBBS and OLIVER
HEAVISIDENEAR.
The end of the 19th century ,and most of the nation and terminology was
established by Gibbs and Edwin Bidwell Wilson in their 1901 book, vector
analysis.
In the conventional form using cross product, vector calculus does not
generalize to higher dimensions.
While the alternative approach of geometric algebra, which uses exterior
products does generalize as discussed below.
3. VELOCITY AND ACCELERATION
VELOCITY
V=dr/dt where v the velocity . It is a
vector function of a scalar variable ‘t’.
Example:
r=4costi +4sintj+6tk
v=dr/dt
v=-4sinti+4costj+6k
ACCELERATION
Acceleration is the rate of change of velocity
a=dv/dt
=d/dt(dr/dt)
Example:
v=-4sinti+4costj+6tk
a=dv/dt
A=-4costi-4sintj+ok
4. GRADIENT
Gradients of a scalar function. Let ф(x,y,z) be a scalar point function. Defiend in a certain
region of space .then the vector point function given by,
gradф= 𝑖 Әф/Әx+jӘф/Әy+kӘф/Әz
is defined as gradient of ф (or) gradф
UNIT NORMAL VECTOR
The vector normal to the surface of ф is
N^= GRAD Փ/|GRADФ|
5. COMPONENT OF VELOCITY AND ACCELERATION
COMPONENT OF VELOCITY
Ā/|A|.V
z
COMPONENT OF ACCELERATION
Ā/|A|.a
operation notation description Notational analogy Domain/range
Gradient Grad(f)=▼f Mesure the rate and
direction of change in
a scalar field.
Scalar multiplication Maps scalar fields to
vector fields
Divergence Div(f)=▼.f Measures the scalar
of a source or sink at
a given point in a
vector field.
Dot product Maps vector fields to
scalar fields
Curl Curl(f)=▼x F Measures the
tendency to rotate
about a point in a
vector field in R
Cross product Maps vector fields to
(pseudo) vector fields
.
6. FORMULA
Scalar potential ( find ф value) GIVEN ▼Փ
Directional derivative Grad ф.a^
Maximum value of directional derivative |grad ф|
Solenoidal vector (div f)=0 (or) ▼.F=0
Irrotational vector Curl F=0 (or) ▼x F=0
7. LAPLACIAN OPERATORS
Operation Notation description Domain/Range
LAPLACIAN ∆f=▼²f=▼.▼f Measures the difference
between the value of the
scalar field with its average
on infinitesimal balls.
Maps between scalar fields
Vector laplacian ▼²=▼(▼.F) - ▼
X(▼XF)
Measure the difference
between the value of the
vector field with its average
on infinitesimal balls.
Maps between vector
fields.
