What is the Need for Vectors?

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.

In math, calculus means the study of the changes, and a vector means a physical quantity that has a magnitude as well as a fixed direction.

What are Vectors?

Before starting vector calculus, let us recall some principles of vector algebra.

(1) Representation

A vector quantity is represented by a directed line segment over the letter.

(2) Addition of Vectors

If there are two vectors represented by the two sides of a triangle, taken in order, then their sum is represented by the third side of the triangle taken in the reverse order.

(3) Dot Product

If there are two vectors and θ is the angle between them, then their dot product is given by:

In rectangular components: 

If $\overrightarrow{a}=〈a_1,a_2,a_3〉$ and $\overrightarrow{b}=〈b_1,b_2,b_3〉$, then $\overrightarrow{a}·\overrightarrow{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3$.

From this, it can be said that the dot product of two vectors is the scalar quantity and it is also called the scalar product.

Note: If two vectors are perpendicular then their dot product is zero as the angle between them is 90^º.

(4) Cross Product

The cross-product of two vectors $\overrightarrow{a}=〈a_1,a_2,a_3〉$ and $\overrightarrow{b}=〈b_1,b_2,b_3〉$ is given by:

$\overrightarrow{a}\times \overrightarrow{b}=〈a_2{b}_3-a_3{b}_2,a_3{b}_1-a_1{b}_3,a_1{b}_2-a_2{b}_1〉$

Or

Where $\widehat{n}$ is the unit vector that is perpendicular to both vectors.

In rectangular coordinates, for two vectors: $\overrightarrow{a}=〈a_1,a_2,a_3〉$ and $\overrightarrow{b}=〈b_1,b_2,b_3〉$.

The cross-product is also equal to:

Note: If two vectors are parallel then their cross product is zero as the angle between them is 0^º.

The above are some basic principles that must be known to learn vector calculus.

What is Vector Calculus?

Vector calculus is also known as vector analysis. It is a branch of mathematics that deals with the differentiation, partial derivatives, line integrals, and integration of vector functions in a three-dimensional space. It plays an important role in the study of partial differential equations and differential geometry. 

Differentiation of a Vector Function

Let us suppose a vector function is represented as shown below: 

The derivative of the function with respect to scalar variable t is written as$\frac{d\overrightarrow{r}}{dt}$and it is equal to the expression shown below:

However, it is not easy to calculate the derivative of the complicated functions using the above limit formula. To evaluate the differentiation of complicated functions in vector calculus, there are some standard rules of differentiation. 

Rules for Differentiation

(1) Sum/Difference Rule:

(2) Product Rules:

$\frac{d}{dt}\left(\overrightarrow{u}·\overrightarrow{v}\right)={\overrightarrow{u}}^{\prime }·\overrightarrow{v}+\overrightarrow{u}·{\overrightarrow{v}}^{\prime }$

$\frac{d}{dt}\left(\overrightarrow{u}\times \overrightarrow{v}\right)={\overrightarrow{u}}^{\prime }\times \overrightarrow{v}+\overrightarrow{u}\times {\overrightarrow{v}}^{\prime }$

(3) Constant Multiplication Rule:

(4) Triple Product Rule:

$\frac{d}{dt}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=[\frac{\overrightarrow{da}}{dt}\overrightarrow{b}\overrightarrow{c}]+[\overrightarrow{a}\frac{\overrightarrow{db}}{dt}\overrightarrow{c}]+[\overrightarrow{a}\overrightarrow{b}\frac{\overrightarrow{dc}}{dt}]$

$\frac{d}{dt}\left[\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})\right]=\frac{\overrightarrow{da}}{dt}\times (\overrightarrow{b}\times \overrightarrow{c})+\overrightarrow{a}\times (\frac{\overrightarrow{db}}{dt}\times \overrightarrow{c})+\overrightarrow{a}\times (\overrightarrow{b}\times \frac{\overrightarrow{dc}}{dt})$

Note:

1. The derivative of a constant vector is zero.

2. If there is a vector having a constant magnitude but variable direction, then $\overrightarrow{r}.\frac{\overrightarrow{dr}}{dt}=0$.

3. If there is any vector having a constant direction but variable magnitude, then $\overrightarrow{r}\times \frac{\overrightarrow{dr}}{dt}=0$.

Velocity and Acceleration

In vector calculus, the velocity of the moving particle is given by the first derivative of the position of the particle with respect to time, and the acceleration is given by the second derivative of the position of the particle or the first derivative of the velocity of the moving particle.

If $\overrightarrow{r}$ denotes the position of a particle then the velocity of the particle is given by the first derivative of the position of the particle:

The acceleration of the particle is given by the first derivative of velocity of particle or second derivative of the position of the particle:

Gradient of a Scalar Field

Let us suppose a function defining a scalar field as $\varphi (x,y,z)$. Then, the gradient of the scalar field is denoted by $grad\varphi$ and it is equal to:

Note:

$\hat{i}\frac{\partial }{\partial x}+\hat{j}\frac{\partial }{\partial y}+\hat{k}\frac{\partial }{\partial z}$ is known as the del operator or also known as nabla operator. Thus $grad\varphi =\nabla \varphi$.

Properties of Gradient

1. For any contact scalar point function $(\varphi )$ the gradient of the function is zero that is $\nabla \varphi =0$.

2. For two scalar point functions:

$\nabla ({\varphi }_1\pm {\varphi }_2)=\nabla {\varphi }_1\pm \nabla {\varphi }_2$

$\nabla (c_1{\varphi }_1+c_2{\varphi }_2)=c_1\nabla {\varphi }_1+c_2\nabla {\varphi }_2$

$\nabla ({\varphi }_1{\varphi }_2)={\varphi }_1\nabla {\varphi }_2+{\varphi }_2\nabla {\varphi }_1$

$\nabla (\frac{f_1}{f_2})=\frac{f_2\nabla f_1+{\varphi }_1\nabla f_2}{f_2{}^2}$

Divergence of a Vector Field

The divergence of a differentiable vector field is defined as:

div V → =∇. V → =( i ^ ∂ ∂x + j ^ ∂ ∂y + k ^ ∂ ∂z ).( V 1 i ^ + V 2 j ^ + V 3 k ^ )$=\frac{\partial V_1}{\partial x}+\frac{\partial V_2}{\partial y}+\frac{\partial V_3}{\partial z}$

Curl of a Vector Field

The curl of a differentiable vector field is defined as:

Properties of Divergence and Curl

1. For any constant: $curl\overrightarrow{a}=0$.

2. $div(\overrightarrow{A}+\overrightarrow{B})=div\overrightarrow{A}+div\overrightarrow{B}$.

3. $curl(\overrightarrow{A}+\overrightarrow{B})=curl\overrightarrow{A}+curl\overrightarrow{B}$.

4. If$\varphi$ is any scalar function:

$div(\varphi \overrightarrow{V})=\varphi (div\overrightarrow{V})+(grad\varphi ).\overrightarrow{V}$

$curl(\varphi \overrightarrow{V})=(grad\varphi \overrightarrow{)\times V}+\varphi curl\overrightarrow{V}$

Integration of Vector Functions

Here the function g(t) is known as the indefinite integral of $f(t)$.

${\displaystyle {\int }_a^b\overrightarrow{f(t)}dt=}{\left.\overrightarrow{g(t)}\right|}_a^b=g(b)-g(a)$

Note: In math, integral calculus is the reverse process of differential calculus. 

Line Integrals

If an integral is evaluated along any curve then the integral is known as a line integral. 

Work Done by a Force

If there is a force F acting on a particle that is moving along an arc AB of any curve P, then the total work done during the displacement from A to B is given by:

Surface Integral

If an integral is evaluated along any surface then it is called the surface integral. If there is a continuous vector function F over any surface S, then the surface integral is written as:${\displaystyle \underset{S}{\iint }\overrightarrow{F}.d\overrightarrow{S}}$

Volume Integral

An integral that is evaluated over a three-dimensional plane is known as a volume integral in mathematics. 

Divergence Theorem

The divergence theorem is one of the most important theorems of vector calculus. The divergence theorem is also known by the name Gauss's theorem. This theorem describes a relation between the surface and volume integral. According to this theorem, if there is a function $\overrightarrow{F}=P\hat{i}+Q\hat{j}+R\hat{k}$, bounded by a closed surface, then:

Where $\hat{n}$ is the unit vector that is normal to the surface S.

In terms of coordinate system:

Green's Theorem

Now, when talking about the vector calculus, then Green’s Theorem is actually the line integrals of the closed curve. Another important theorem of vector calculus is Green's theorem.

Green's theorem states that if there are two continuous functions $M(x,y)$and $N(x,y)$ in a region D bounded by any closed curve S, then:

Partial Derivatives

The partial derivatives are used to differentiate a vector function that has more than one variable. With the partial derivatives, one can find the slope or the rate of change of the vector function with respect to any one variable. So, by using the partial derivatives it is seen that the vector functions might change differently for the different variables.

Formulas

Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• M.Sc. Mathematics
• B.Sc. Mathematics