Vector calculus is concerned with differentiation and integration of vector fields, primarily in 3D Euclidean space. It plays an important role in differential geometry and studying partial differential equations. Vector fields assign a vector to each point in a space to model things like fluid speed/direction or magnetic/gravitational forces. Vector calculus distinguishes between vector fields, pseudovector fields, scalar fields, and pseudoscalar fields.

1. VECTOR CALCULUS
BY,
J. MANJULA
ASSISTANT PROFESSOR OF MATHEMATICS
BON SECOURS COLLEGE FOR WOMEN

2. INTRODUCTION
 Vector calculus, or vector analysis, is concerned with
differentiation and integration of vector fields, primarily in 3-
dimensional Euclidean space.
 The term “vector calculus” is sometimes used as a synonym for
the broader subject of multivariable calculus, which includes
vector calculus as well as partial differentiation and multi
integration.
 Vector calculus plays an important role in differential geometry
and in the study of partial differential equation.

3. BASIC OBJECTS
Scalar fields
 A scalar field associates a scalar value to every point in a
space.
 The scalar may either be a mathematical number or a
physical quantity.
 Examples of scalar fields in applications include
• the temperature distribution throughout space,
• the pressure distribution in a fluid,
• spin-zero quantum fields, such as the Higgs field.

4. Vector fields
A vector field is an assignment of a vector to each point in a subset
of space.
 Vector fields are often used to model.
 for example, the speed and direction of a moving fluid throughout
space, or the strength and direction of some force, such as the
magnetic or gravitational force, as it changes from point to point.

5. Vectors and pseudovectors
 In more advanced treatments, one further distinguishes
pseudovector fields and pseudoscalar fields, which are identical to
vector fields and scalar fields .
for example, the curl of a vector field is a pseudovector field, and if
one reflects a vector field, the curl points in the opposite direction.

6.  An Architect Engineer uses integration in determining the amount of the necessary
materials to construct curved shape constructions and also to measure the weight
of that structure.
Calculus is used to improve the architecture not only of buildings but also of
important infrastructures such as bridges.
 In Electrical Engineering, Calculus is used to determine the exact length of power
cable needed to connect two substations.
 Space flight engineers frequently use calculus when planning for long missions

10.  Biologists use differential calculus to determine the exact rate of
growth in a bacterial culture when different variables such as
temperature and food source are changed.
 It is used to determine the rate of a chemical reaction and to
determine some necessary information of Radioactive decay
reaction

11.  Statisticians use calculus to evaluate survey data to help develop
business plans for different companies.
 An operations research analyst will use calculus when observing
different processes at a manufacturing corporation.
 They can help a company improve operating efficiency, increase
production, and raise profits.

12. A graphics artist uses calculus to determine how different three-
dimensional models will behave when subjected to rapidly changing
conditions.
It can create a realistic environment for movies or video games.

13. THANK YOU