Vector & scalar point function, vector field, Vector differentiation and Integration

1. Vector Calculus

2. Point function
• A point function u = f(P) is a function that
assigns some number or value u to each point
P of some region R of space.

3. Scalar point function
• 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 is called a
scalar point function
• Examples
• 1. The temperature distribution within some body at a
particular point in time.
• 2. The density distribution within some fluid at a particular
point in time.

4. 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.
• 3. Velocity at different points within a moving fluid.
• 4. Acceleration at different points within a moving fluid

5. 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.

6. Field
• The word ‘field' signifies the variation of a
quantity (whether scalar or vector) with
position

7. 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

8. • 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.

9. Vector differential operator (or) Del-
(𝛁) 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.

10. 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

11. Few Properties of Del 𝛁
• 1. 𝛻∅ is a vector quantity
• 2. 𝛻∅ = 0 ⇒ ∅ is a constant
• 3. 𝛻 ∅1. ∅2 = ∅1 𝛻∅2 + ∅2 𝛻∅1
• 4. 𝛻
∅1
∅2
=
∅2 𝛻∅1− ∅1 𝛻∅2
∅2
2
• 5. 𝛻 ∅ ± 𝜓 = 𝛻∅ ± 𝛻𝜓

12. Directional Derivative
• The derivative of a point function (scalar or vector) in a particular direction is
called the directional derivative of the function in that particular direction.
• (i.e) the rate of change of the function in the particular direction.
•
• (i.e) if 𝑛 is a unit vector, then 𝑛 · 𝛻∅ is called the directional derivative of ∅ in the
direction 𝑛. The directional derivative is the rate of change of ∅ in the direction 𝑛.
• The directional derivative of ∅(𝑥, 𝑦, 𝑧) in the direction of the vector 𝑎 is given by
𝛻∅ .
𝑎
𝑎
• (since
𝑎
𝑎
is the unit vector along 𝑎 )
• The gradient indicates the maximum and minimum values of the directional
derivative at a point.
•
• The directional derivative of ∅ is maximum in the direction of 𝛻∅
• The maximum directional derivative is 𝛻∅ or grad ∅

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

14. Solenoidal vector
• A vector 𝐹 is said to be solenoidal , if div 𝐹 =
0 (i.e) 𝛻. 𝐹 = 0

15. Curl of vector function
• If 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 is a continuously differentiable
vector point function in a given region of space, then
the curl of 𝐹 is defined by 𝛻 𝑋 𝐹 = 𝑐𝑢𝑟𝑙 𝐹 =𝑖 𝑗 𝑘
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐹1 𝐹2 𝐹3
• Curl 𝐹 is also said to be rotation 𝐹
• Note: Curl measures the tendency of the fluid to swirl
around the point. Curl is a vector

16. Irrotational Vector
• A vector 𝐹 is said to be Irrotational, if Curl
𝐹 = 0 (i.e) 𝛻X 𝐹 = 0

17. Scalar potential
• If 𝐹 is irrotational vector, then there exists a
scalar function ∅ such that 𝐹 = 𝛻∅. Such
scalar function ∅ is called scalar potential of 𝐹

18. Conservative
• If 𝐹 is conservative then𝛻𝑋 𝐹 =
0 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐹 = 𝛻∅, where ∅ is scalar
potential

19. Angle between two surfaces
• If ∅1 = 𝑐 & ∅2 = 𝑑 are two given surfaces,
then the angle between these two surfaces is
given by 𝐶𝑜𝑠 𝜃 =
𝛻∅1.𝛻∅2
𝛻∅1 |𝛻∅2|
where 𝜃 is the
angle between given two surfaces
• Note: if the surfaces are orthogonal, then
𝛻∅1. 𝛻∅2 = 0 (since 𝜃 =
𝜋
2
)

20. Vector Integral
Line Integral
• 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
• Let 𝐹 = 𝐹1 𝑖 + 𝐹2 𝑗 + 𝐹3 𝑘 be a vector 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
• This is also called tangential line integral of 𝐹 over C
• The limit of the integral is the corresponding values of the end
points of the curve C

21. Circulation
• If C is a simple closed curve, then the line
integral over C, 𝐶
𝐹 . 𝑑 𝑟 is called circulation of
𝐹 over C.

22. Work done by a force
• If 𝐹 is a force acting on a particle which is
moving along the given curve C, then the work
done by the force is given by 𝐶
𝐹 . 𝑑 𝑟

23. Surface Integral
• 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, then the integral is
𝑆
𝐹. 𝑛 𝑑𝑆
• 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|

24. 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

25. Integral Theorem

26. Green’s 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
𝑑𝑥𝑑𝑦 , where C
is the positive oriented closed curve
• ( i.e C is in anti-clock wise direction)
• Note : By Green’s theorem , Area of region =
1
2
𝑥𝑑𝑦 − 𝑦𝑑𝑥

27. 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.

28. 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