Line integrals
Hello Students
Today we will learn about Line Integrals
Line Integral is the part of complex analysis
We have talked about complex analysis in the previous 3 sessions.
First was about basic vector point function, what is a vector, etc.
After that, we learned about vector differentiation, direction derivative, etc
and we learned about the diversion concept
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so that you understand this video
more precisely.
So let us see,
What is the Line Integral?
Line Integral refers to the normal Integration
but vector has one direction
integration is done using the 1D concept.
Like in Double Integration we use the 2D concept,
triple integration uses a 3D concept and so on.
So let us see how we solve the line integral in vector.
Also, let me tell you
in complex analysis, vector analysis, numerical analysis
in all these, we study 2 things
One is Differentiation and other is the Integration
We are going to study Integration.
Let us see how to easily solve in integration.
Previously we discussed how to solve complex analysis
in an easy way.
Similarly, we can solve Line Integral
using Green Theorem, Stoke Theorem
by converting the integral into the surface.
We will understand it later.
We will also understand its applications
in the next video.
that by using Line Integral what all we can solve.
For eg: we can calculate work done, force
and other things, that we will discuss in our next video.
So let us see what is Line Integral?
Let f(x,y,z) be a vector function and curve AB
Line Integral of a vector function F
along the curve AB
is defined as the line integral
of the component of F vector
along the tangent
to the curve AB
or else in another language we can say
The Line Integral which is to be evaluated
along the curve is called Line Integral.
Line Integral means that you have to solve
it on a particular path.
For eg: Given a line with two points A and B
from here to here
whose coordinates are (0,0) and (4,4)
and we want to solve it.
We need to solve the Integration along this line.
We will be given a path along which we have to solve.
Let us understand it with a question
that how we will solve it.
Here is the question.
 
 
 
 
 
 
 
 
 
 
 
 
let us see how to solve it
First of all, this is the way we write line integral.
where, r = xi + yj + zk
and dr = dxi + dyj + dzk
Let's see how we will solve it.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
This is a line integral and it should always be in one variable.
either in X or Y. Then only we can solve it.
In the question, there must be given a relationship
 
 
 
 
 
 
So what we will do.
Either we will convert this whole integration  in terms of x or y
 
 
 
 
The expression is converted in terms of x
and the limit of x is from (0,0) to (4,4)
 
 
 
 
 
 
 
 
 
This is how we will solve Line Integral.
Let us solve more 1 or 2 questions.
Let us take another question.
 
 
 
 
We have to calculate along this line.
this is the x-axis, this is the y-axis
so x=1 is this line and x=-1 is this line.
so y=1 is this line and y=-1 is this line.
First of all, we need to trace
write down the coordinates of encircled points.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
The upcoming videos will be on green and stoke theorem
This question is going to be lengthy
but using green and stoke theorem we will solve it in less time.
I will teach that in detail in the next video
but sometimes there is a question in exams,
Verify Green's Theorem or Stokes Theorem?
so here we should be aware of the concept.
we need to calculate the value of f.dr
but we should be aware of the path.
we will start from here and then continue.
and then we will add four of them.
 
 
 
 
 
 
 
Understand the trick carefully.
wherever there is dx write i instead.
 
wherever there is dy write j instead.
 
 
We are looking along AB
In this line AB, we have coordinate A and B
and here x is  not changing
 
x is constant so its differentiation will be 0.
and y is going from -1 to +1
 
 
 
 
 
 
 
 
 
the limit will be -1 to +1 as y is changing and x is constant.
now let us solve.
this is an even function. So we will do it twice.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
now we will solve from B to C
here y is constant
so its dy will be 0
x is changing going from -1 to +1
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I will reverse the limits and will put minus sign.
or else we can take minus common.
 
 
this is an odd function
and this is even.
we will do it twice.
 
 
 
 
Now we will see along CD
in C to D, x is constant i.e x=-1
so dx=0
y is changing
 
y is from +1 to -1
This thing is to be kept in mind.
 
 
This is the function given
we will 0 in place of dx
and in place of x, we will keep -1
 
 
 
 
 
 
 
 
 
 
 
 
Now we will take along DA
 
 
here y is constant i.e y=-1
and dy=0
and x is from -1 to +1
This needs to be kept in mind.
 
 
here in place of dy it will be 0.
and in place of y, it will be -1
 
 
 
 
this is an odd function and its integration will be 0.
 
 
 
 
Now, the values of all the above
will be put here in equation 1
Let me solve it here.
 
 
 
 
along AB the value is 8/3
along BC the value is 2/3
along CD the value is -8/3
and along DA the value is 2/3
 
I think there is some problem
Here it should be minus
 
*solve the equation*
This is the way how we solve a particular equation.
Now in exams, it is generally asked
that solve this question by green theorem
solve this by stoke theorem
Using green theorem or stoke theorem we can solve this in just 3 to 4 lines.
 
so you should also know that and sometimes it is also asked to verify
So you need to solve through this method also and by using the theorem
and both the answers should be zero.
Let us finish this topic with the last two questions.
So these are the two questions
on how we can solve this with the line integral concept.
 
 
 
 
 
What we have to do is?
 
wherever there is i write dx
after doing dot product, this is the final expression
 
let us simplify it
 
 
 
 
 
so this is the value we are getting
look at these two terms
 
what do you observe?
here sin y is constant
we will do differentiation with respect to x
 
we will merge these two and we can write
 
 
when we will open the brackets by using u*v
 
 
 
 
we have done differentiation with respect to x
 
 
 
 
so this is a circle
we will take its coordinates as shown.
also, the coordinates satisfy this equation.
 
 
 
Here, theta is from 0 to 2pi
 
 
 
In place of x and y, we will write as shown
 
 
 
 
 
as it is dy we will differentiate it here.
 
 
 
 
 
 
 
 
 
 
 
 
Now wherever there is theta, we will put 2pi
 
 
 
Ultimately after solving, we will get value as 0
 
 
 
 
 
 
 
 
*solve the equation as shown*
 
 
 
 
 
 
 
So our final is as shown here.
This is the way how we can solve it.
We can solve this using green and stoke theorem which we will discuss later.
Let us see our last question.
 
 
 
 
And here F is given as
Let us see how we will solve it
 
 
 
 
wherever j is coming write dy
 
What the question is saying that
you have to calculate from (0,0,0) to (1,0,0)
from (1,0,0) to (1,1,0)
and from (1,1,0) to (1,1,1)
Let us consider this as A, B, C and D
 
 
 
 
 
 
 
First we will calculate along AB
 
 
 
 
 
 
in AB x is changing from 0 to 1
and y is 0 so dy=0
and z is also 0 so dz=0
there is no change in y and z
 
 
x limit will be from 0 to 1
wherever there is dy and dz put 0
and in place of y put 0
then we are left with..
 
Now will calculate along BC
 
 
in B to C, y is changing
x and z is constant
 
x=1 so dx=0
y= 0 to 1
z=0 so dz=0
 
the limit is from 0 to 1
wherever there is dx we will put 0
and also dz=0
so required value is 0
Now will calculate along CD
 
 
here x=1 so dx=0
y=1 so dy=0
and z is changing from 0 to 1
 
 
 
the limit will be 0 to 1
here dx and dy will be 0
and in place of x put 1
required value is as shown
 
 
All these three values will be substituted in equation 1
Required answer is
 
 
 
 
So students
Today we learned about vector calculation
and how we can solve line integral
I hope you all understood it really well
The next topic we will going to discuss is
Applications of line integrals
after that, we will learn about green theorem, stoke theorem.
 
Check out my previous videos on divergent,  irrotational vector etc.
also other videos related to engineering and BSC students
on the topics like integral transform, complex analysis, partial differential equation
Linear algebra
 
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