1. Vector calculus deals with vector-valued functions and their derivatives. It includes vector point functions that assign vectors to points in space, as well as scalar point functions that assign real numbers. 2. Key concepts include the gradient of a scalar function, the divergence and curl of a vector function, and vector fields to describe variations of quantities like velocity over a region of space. 3. Vector calculus theorems relate integrals over surfaces to integrals over bounding curves or volumes, such as Green's theorem, Stokes' theorem, and the divergence theorem. These allow problems involving surfaces and volumes to be solved via line and surface integrals.

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