 
 
 
 
Hello students, today I am here with a new topic
and the topic is the application of line integral
before this, I have taught line integral to you
if you have not watched that video then you can go and press the 'I' tab
and go through to my old videos  here I have told
what is gradient, curl
concept of diversion
how to find the derivative of direction
these all 2-3 old video you can go through
and watch that video then understand these this concept
then there will be no difficulties in learning so today we will
discuss that application of line integral
so 1st application is work done
 
 
 
 
 
 
1 note is there that at the place 'F ' it is a velocity
then we called them as circulation of
'F'
around curve c
2nd application is independent  of the path
if there is some force and it is not
dependent on its path
how to find this if it is an irrotational
vector or it is a conservative irrotational vector field
so directly we can do its independent path
if we solve this  and directly we will use its limit
and we do its integration
so I will explain to you with a question without that you will not get that
let's see the question
 
 
what is the concept of the scalar potential
let's take an example
 
 
if a take a gradient of this scalar
students if you don't know what is a gradient then you can go
then click on 'I'  tab and go
through my old video their I have explained about what is a gradient
here I will not able to explain the concept of gradient
 
 
 
 
if I take its curl
then its curl will be 0
gradient curl is 0
now what is the scalar potential
this is a scalar
and we have taken its gradient then it became  a vector
this type of vector is given
now the question may be asked that this  vector
is of which
scalar, means whose gradient we have taken
and it became a vector
we have to find F
this F is called a scalar potential
this f is made of a function of
a gradient
 
this need to be understand
 
 
 
 
simply I will explain to you that
1st you have to prove that its curl should be 0
that it is rotational or conservative vector field
if it is rotational then we can find its scalar potential
means we can cancel out its gradient
then we will fin ita work done
so we will solve a question
 
 
 
 
 
 
now we will find its curl
 
 
 
 
on this topic, I have already uploaded one video
here you can click on 'I' tab and go through the video of curl
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
this f is irrotational
 
 
 
 
 
if this is conservative vector field then we can find its scalar potential
 
 
 
 
 
 
means we have to do its integration
a gradient is a derivative
cancel out the gradient means
doing its integration
I am telling you a short trick
 
 
 
 
 
this you can do in exam also
 
 
 
while doing integration variable should be
separated
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
this is scalar potential
 
if you want to know that its is correct or not then you can take its gradient
 
 
now we have to find its work done
work done means doing its integration
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
like these, we will solve this question
let's take one more question
same question as to the previous one
an extra thing is that conservative vector field is given
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
one last question on an independent path
 
 
 
 
 
if we have to prove that
it is independent of the path
its is an irrotational or conservative vector field
if it is irrotational or conservative vector filed
means its integral
is independent of its path
 
 
same as the previous question nothing new
the only language is changed
 
 
I have already told this if anyone doesn't know
before this go and watch line integral
then you will get how to solve these
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
like this way we will solve this question
I have explained to you that what is an application of line integral
my video sare coming on large scale
 
 
 
 
 
all this video I have uploaded
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thank you so much for watching my video
