Point • Point function u = f(P) is a function that

assigns some number or value u to each

function point P of some region R of space.
 • A scalar point function is a function that
assigns a real number (i.e. a scalar) to each
point of some region of space. If to each point
(x, y, z) of a region R in space there is
assigned a real number u = Φ(x, y, z), then Φ
Scalar point is called a scalar point function
function Examples:
1. The temperature distribution within
somebody at a particular point in time.
2. The density distribution within some fluid at
a particular point in time.
 Vector point function
• A vector point function is a function that
assigns a vector to each point of some region
of space. If to each point (x, y, z) of a region
R in space there is assigned a vector F = F(x,
y, z), then F is called a vector point function.
Such a function would have a representation
•
• Examples
• 1. Gravitational field of the earth.
• 2. Electric field about a current-carrying wire.
• 3. Magnetic field generated by a magnet.
• 4. Velocity at different points within a
moving fluid.
• 5. Acceleration at different points within a
moving fluid
• Example of scalar and vector point
functions
• Consider a cactus, with long pointed thorns
over it. The presence or absence of a thorn
at a particular location (x,y,z) on the cactus
is a scalar point function; the function takes
values 1 or 0, depending upon whether the
thorn is present or not at location (x,y,z).
• Consider another function: direction of the
thorns. In this case, as a function of
location ( x, y, z) on the cactus you get a
vector point function represented by the
vector in which the thorn is pointing.
 Field
The word ‘field' signifies
the variation of a
quantity (whether scalar
or vector) with position
 Scalar Field
• A scalar field is a function that gives us a single value
of some variable for every point in space. (i.e) a scalar
field associates a scalar value to every point in a space
• Scalar field- where the quantity whose variation is
discussed is a scalar. For example - pressure,
temperature are scalar fields since they do not have
any direction. Example:
• Atmospheric temperature variation as a function of
altitude above the
• Earth’s surface
 Vector Field
A vector is a quantity which has both a magnitude and a
direction in space. Vectors are used to describe physical
quantities such as velocity, momentum, acceleration and force,
associated with an object. However, when we try to describe a
system which consists of a large number of objects (e.g., moving
water, snow, rain,…) we need to assign a vector to each
individual object.

Vector field- where the quantity whose

variation is discussed is a vector. For example,
electric field, magnetic field, gravitational field
etc.
• Example:
• As snow falls, each snowflake moves in a specific
direction. The motion of the snowflakes can be analyzed
by taking a series of photographs. At any instant in time,
we can assign, to each snowflake, a velocity vector which
characterizes its movement. The falling snow is an
example of a collection of discrete bodies.
• Another example if we try to analyze the motion of
continuous bodies such as fluids, a velocity vector then
needs to be assigned to every point in the fluid at any
instant in time. Each vector describes the direction and
magnitude of the velocity at a particular point and time.
The collection of all the velocity vectors is called the
velocity vector field.
• The gravitational field of the Earth is another example of a
vector field which can be used to describe the interaction
between a massive object and the Earth.
• An important distinction between a vector field and a
scalar field is that the former contains information about
both the direction and the magnitude at every point in
space, while only a single variable is specified for the
latter.
 (or) Del- (𝛁) operator
Vector differential operator

• The Del operator is defined as

follows • 𝛻 = 𝑖 𝜕 𝜕𝑥 + 𝑗 𝜕 𝜕𝑦
+ 𝑘 𝜕 𝜕𝑧 • 𝛻2 = 𝛻. 𝛻 = 𝜕2 𝜕𝑥2
+ 𝜕2 𝜕𝑦2 + 𝜕2 𝜕𝑧2 , Which is
called Laplacian operator.
 The gradient of a scalar function
(grad)
• The gradient is a derivative (or) rate of change of a multi
variable function, which has component for each direction.
• If ∅ 𝑥, 𝑦, 𝑧 is a scalar point function continuously
differentiable in a given region of space, then the gradient of
∅ is defined as
• 𝛻∅ = 𝑖 𝜕∅ 𝜕𝑥 + 𝑗 𝜕∅ 𝜕𝑦 + 𝑘 𝜕∅ 𝜕𝑧
• It is denoted by grad ∅ = 𝛻∅ (or) grad ∅ = 𝑖 𝜕∅ 𝜕𝑥
 • 1. 𝛻∅ is a vector quantity
Few • 2. 𝛻∅ = 0 ⇒ ∅ is a constant
• 3. 𝛻 ∅1. ∅2 = ∅1 𝛻∅2 + ∅2 𝛻∅1
Del 𝛁
Properties of
• 4. 𝛻 ∅1 ∅2 = ∅2 𝛻∅1− ∅1 𝛻∅2 ∅2 2
• 5. 𝛻 ∅ ± 𝜓 = 𝛻∅ ± 𝛻𝜓
 Directional Derivative
(i.e) if 𝑛 is a unit vector,
then 𝑛 · 𝛻∅ is called the
The derivative of a point
function (scalar or vector)

in the direction 𝑛. The

in a particular direction is (i.e) the rate of change of directional derivative of ∅
called the directional the function in the
derivative of the function particular direction. directional derivative is

the direction 𝑛.
in that particular the rate of change of ∅ in
direction.

of ∅(𝑥, 𝑦, 𝑧) in the
The directional derivative
The gradient indicates
direction of the vector 𝑎
the maximum and The directional derivative
is given by 𝛻∅ . 𝑎 𝑎
direction of 𝛻∅
minimum values of the of ∅ is maximum in the
(since 𝑎 𝑎 is the unit
directional derivative at a
vector along 𝑎 )
point.

derivative is 𝛻∅ or grad
The maximum directional

∅
 Divergence of a vector function
• If 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 is a continuously differentiable
divergence of 𝐹 is defined by 𝛻. 𝐹 = 𝑑𝑖𝑣 𝐹 = 𝜕 𝜕𝑥 𝑖 +
vector point function in a given region of space, then the
Solenoidal vector

A vector 𝐹 is said to
be solenoidal , if div 𝐹
= 0 (i.e) 𝛻. 𝐹 = 0
 If 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 is a continuously

region of space, then the curl of 𝐹 is defined by

differentiable vector point function in a given

𝛻 𝑋 𝐹 = 𝑐𝑢𝑟𝑙 𝐹 =𝑖 𝑗 𝑘 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝐹1 𝐹2 𝐹3

Curl of
Curl 𝐹 is also said to be rotation 𝐹
vector
function

Note: Curl measures the tendency of the fluid to

swirl around the point. Curl is a vector
 A vector 𝐹 is said to
Irrotational
Curl 𝐹 = 0 (i.e)
be Irrotational, if

𝛻X 𝐹 = 0
Vector
Scalar • If 𝐹 is irrotational vector, then there exists a scalar
function ∅ such that 𝐹 = 𝛻∅. Such scalar function
potential ∅ is called scalar potential of 𝐹
 • If 𝐹 is conservative then𝛻𝑋 𝐹 = 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐹
= 𝛻∅, where ∅ is scalar potentialIf 𝐹 is
Conservati conservative then𝛻𝑋 𝐹 = 0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐹 = 𝛻∅,
where ∅ is scalar potential
ve
 • If ∅1 = 𝑐 & ∅2 = 𝑑 are two given

two surfaces is given by 𝐶𝑜𝑠 𝜃 =

Angle surfaces, then the angle between these

between two 𝛻∅1.𝛻∅2 𝛻∅1 |𝛻∅2| where 𝜃 is the

angle between given two surfaces
surfaces •
then 𝛻∅1. 𝛻∅2 = 0 (since 𝜃 = 𝜋 2 )
Note: if the surfaces are orthogonal,
 • An integral evaluated over a curve is called
line integral
Let C be the given curve and 𝑟 = 𝑥 𝑖 + 𝑦 𝑗
+ 𝑧 𝑘 be the position vector of any point on
•
C
Vector • Let 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a vector
Integral Line point function defined on C
The line integral of 𝐹 over C is defined by 𝐶
𝐹 . 𝑑 𝑟 , where 𝑑 𝑟 = 𝑑𝑥 𝑖 + 𝑑𝑦 𝑗 + 𝑑𝑧 𝑘
•
Integral
of 𝐹 over C
• This is also called tangential line integral

• The limit of the integral is the corresponding

values of the end points of the curve C
•
integral over C, 𝐶 𝐹 . 𝑑 𝑟 is called
If C is a simple closed curve, then the line

circulation of 𝐹 over C.
Circulation
 • If 𝐹 is a force acting on a particle which
Work done is moving along the given curve C, then

by 𝐶 𝐹 . 𝑑 𝑟
the work done by the force is given
by a force
• An integral evaluated over a surface is
called surface integral.
• Let S be the given surface and 𝐹(𝑥, 𝑦, 𝑧)
be the vector point function defined on
each point of the surface S.
• The flux integral is defined by 𝑆 𝐹. 𝑑 𝑆
• If 𝑛 is the unit normal to the surface S,
Surface
then the integral is 𝑆 𝐹. 𝑛 𝑑𝑆
Integral
• If 𝑑𝑠 is the small element of the surface S
and 𝑘 is the unit normal to the xy-plan
and 𝑑𝑥𝑑𝑦 is the projection of the
element 𝑑𝑠 on xy- plane , then the surface
integral/ flux integral s defined by
• 𝑆 𝐹. 𝑛 𝑑𝑆 = 𝑅 𝐹. 𝑛 𝑑𝑥𝑑𝑦 | 𝑛. 𝑘|
 Volume Integral
An integral evaluated over a volume bounded by a surface is
called volume integral.

If V is the volume bounded by a surface S and 𝐹 = 𝐹1 𝑖 +

𝐹2 𝑗 + 𝐹3 𝑘 is a vector field over S

Then the volume integral over V is defined by

𝑉 𝐹 𝑑𝑉 = 𝑖 𝑉 𝐹1 𝑑𝑉 + 𝑗 𝑉 𝐹2 𝑑𝑉 + 𝑘 𝑉 𝐹2 𝑑𝑉
Integral Theorem
 • If M(x,y) and N(x,y) are two multi variable
continuous and differentiable functions in the
given region R on a surface then
• 𝐶 𝑀𝑑𝑥 + 𝑁𝑑𝑦 = 𝑅 𝜕𝑁 𝜕𝑥 − 𝜕𝑀 𝜕𝑦
Green’s 𝑑𝑥𝑑𝑦 , where C is the positive oriented
Theorem closed curve
• ( i.e C is in anti-clock wise direction)
•
1 2 𝑥𝑑𝑦 − 𝑦𝑑𝑥
Note : By Green’s theorem , Area of region =
 Stoke’s Theorem

• If a vector function 𝐹 is continuous

and has continuous partial derivative
in an open surface bounded by a
simple closed curve C, then
• 𝑆 𝛻𝑋 𝐹 . 𝑛 𝑑𝑆 = 𝐶 𝐹 . 𝑑 𝑟 ,
where 𝑛 is the unit normal to the
surface.
 Gauss Divergence Theorem

• If a vector function 𝐹 is continuous and has continuous partial

derivative in the volume V bounded by a closed surface S,
then
• 𝑆 𝐹. 𝑛𝑑𝑆 = 𝑉 𝛻. 𝐹 𝑑𝑉 , where 𝑛 is the unit normal drawn
outward to dS