Electromagnetic Field Theory

Vector Calculus

TWO or THREE DIMENSIONAL PROBLEM = VECTOR

One Dimensional Problem = Scalar

🌾As we discussed earlier in the topic, ‘One dimensional problem = Scalar’, 

  • Quantities with NO direction = SCALAR
  • Quantities with only ONE direction = SCALAR
  • Quantities with only TWO (opposite) directions = SCALAR

Quantities that vary only in ONE DIMENSION are termed as SCALAR.

Two or Three Dimension:

  • NO direction = No dimension
  • Only ONE direction = ONE dimension $(+\;x-axis)$
  • Only TWO direction = ONE dimension $(\pm\;x-axis)$
  • More than TWO direction = TWO or THREE dimension

In general, every possible direction involves in a TWO dimensional space can be defined using the following four directions$$North-South-East-West$$Mathematically we define it using $$\pm x-axis,\pm y-axis$$

Mathematically three dimension can be defined using $$\pm x-axis,\pm y-axis, \pm z-axis$$

More than ONE dimension = Vector:

Location

Post OfficeHospitalTea shop2 km5 km3 km1 km

A Maths teacher in the car needs to go to the Hospital. His wife needs to go to the tea shop and his son to the Post office. None of them speaks English. 

Each of them handed you a piece of paper. One shows a drawing of hospital, other shows a tea shop and the third one shows a post office.

How to guide them only with numbers???

  • If the car goes in the same direction for $5\;km$, he will reach the hospital. So, we can write $+5$ in the paper.
  • If the car turn back and go for $2\;km$, she will reach the tea shop. So, we can write $-2$ in the paper.
  • If the car goes in the same direction for $1\;km$, then take a left turn and travels for $3\;km$, his son will reach the post office. So, we can write $1\;\overrightarrow{a_x}+3\overrightarrow{a_y}$ in the paper.

Whenever, more than two directions involve in a problem, we need to define the direction.

Hence we can say quantities that vary in two or three dimensions can be termed as VECTOR.

Upto ONE dimension = SCALAR.

TWO or THREE dimension = VECTOR.

Need for Vector Calculus:

Assume a bullock pulls a cart $(F_1)$ and a person is pushing the cart from behind $(F_2)$. Now total force in moving the cart is nothing but, the sum of pulling force and pushing force. Right?? $$F=F_1+F_2$$

Why??

Pulling force and pushing force lies in the same direction, only one direction is involved in this problem, we can use scalar calculus here (i.e.) ordinary addition.

Can we apply the same, when forces not acting along a line???

NO

Consider the above problem, here $F_1$, $F_2$ are not acting along the same line, hence we cannot find the resultant force using Scalar calculus (normal addition).

This problem is a “Two dimensional problem”. 

Scalar calculus can work upto ONE dimension. To solve two dimensional problem we need some technique other than scalar calculus.

Newton's Parallelogram law:

Even before the idea of ‘Vector Calculus’, Newton formulated Parallelogram law of forces in order explain the forces acting on a body.

Statement: If two forces, acting at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

This is the first law that considers magnitude as well as direction.

From $16^{th}$ century itself Scientists knew, they need some new area of mathematics to deal this kind of problems.

Birth of Vector Calculus:

In $1901$, based on series of lectures given by Josiah Willard Gibbs, Edwin Bidwell Wilson published a book on vector calculus named, “Vector Analysis: A Text Book for the Use of Students of Mathematics and Physics”.

This is the first text-book ever published on Vector Calculus.

After the publication of this book, the field of mechanics, electrostatics, electromagnetism, etc., adapted Vector calculus.

Vector and Vector Calculus:

Vector – Quantity that has to be defined by both its magnitude and direction.

Vector Calculus is a branch of study,

  • Especially made for quantities that vary in two or three dimension.
  • For one dimensional variation scalar calculus is enough.
  • includes vector addition, subtraction, multiplication (Dot product & Cross product), differentiation (Divergence, Curl), integration (Line, Surface & Volume), etc.,