Hello students, I am here
with the second part on vector calculus
In first we discussed what is vector differentiation
and how we do the differentiation
if you have any function
that is scalar then how we will do its differentiation
 
and we call it as gradient
and if by chance we have a function as a vector
so we have two things
one is dot product and one is the cross product
among which one is called as divergent and other curl
and we also discussed some questions
today I am taking the topic Gradient
and we will discuss some questions
we will study about direction derivative, which is very important
and based on that we get questions in the exam
How we calculate the direction derivative
plus we have a concept here of unit normal vector
how to calculate the normal vector
there is any surface we have to calculate its perpendicular vector is called a normal vector
and there is one more, we have two surfaces and we have to calculate the angle between them
we will discuss those concepts here
so look at the two questions here
I have discussed what is a gradient
if you are given any scalar then its gradient will be
let us understand it with a question
 
 
so what we will do here is
 
 
 
 
 
F is given as
 
it is scalar so it will be multiplied by the whole term
 
 
 
 
 
 
this derivative is partial derivative here
so what we will do is
here x will be differentiated and y and z will be constant
so it will come as
 
 
 
 
 
 
now you are being asked here
you have to find the gradient at (1,1,1)
you will keep the points here
 
put x=1, y=1, and z=1
 
so this is the gradient of F
so we will calculate the gradient of any function this way
so students here is the next question
 
any question is given, first of all, what we will do is
 
 
I should tell you the definition of vector
so first of all what is the normal vector
we denote it by n and its formula is
if we are given any question and we have to calculate the normal vector
so we will calculate the gradient
and if we have to calculate unit normal vector then we will do as follows
so this is the concept we will use here
and we have discussed their definition in the previous video
let me tell you again
so what we will do here is
 
 
 
 
 
 
solve as follows
multiply it with the whole term
here is the definition of the normal vector
 
 
 
 
 
we will differentiate it with respect to x
 
 
 
differentiate it with respect to y
 
 
now substitute the points
 
 
putting the points
 
 
 
 
and our normal vector will be
 
 
 
 
solve as follows
 
 
 
 
 
 
 
so this is the way we calculate gradient and normal vector
and unit normal vector
now we will see how to calculate the direction derivative
so, students, we have two questions here on directional derivative
It's first part video, I have already shared
in that, I discussed the definition
let us recall it
 
directional derivative in direction A has the formula as
 
 
for example, we have a question here
 
 
so first of all what we will do is
 
 
 
 
 
 
solve as follows
 
we will differentiate this with respect to x
so we will get
 
now we will differentiate this with respect to y
 
now differentiating it with respect to z
 
 
so this we got the gradient of F
now it is asking
you have to calculate the directional derivative in A direction
and we have the point as
substituting the points
 
put x=-1, y=-1 and z=-1
 
 
 
 
 
 
we will simplify it
 
now you have to calculate in this direction
 
 
 
 
 
 
solve as follows
 
 
 
 
this is the unit vector and in that, we solve like this
 
you will get the value as
 
 
 
so this is our directional derivative in the direction A
 
let us see one more question
this question has asked one more thing
 
 
 
you have to calculate the maximum value also
so let us begin
first of all what we will do
we will calculate gradient of F
 
 
 
 
 
 
 
 
 
now the point is given as
 
 
 
 
we will calculate the directional derivative of this
 
 
 
 
solve as follows
 
 
 
 
we will simplify it
 
 
 
now we will calculate the maximum directional derivative
 
so students, let me tell you
the maximum directional derivative is calculated when
 
we want to calculate the derivative in the same direction as of directional derivative
 
its maximum directional derivative will be calculated in this direction
derivative in its own direction is called maximum directional derivative
 
so we will calculate it as follows
we have the formula as
 
 
 
so we will get the value as
 
now you will have a question how we got this formula
 
so let me tell you how we got this
 
 
 
 
 
its unit factor will be
when you will simplify it
 
 
 
 
 
so this is the way we get the formula and solve
let us take one last question
when we are given two surfaces, how to calculate the angle between them
with the help of gradient
 
so here i have taken two more questions
when you are given two surfaces  and a point
and you have to calculate the angle
we can do this with the help of gradient
let us see how we will do it
we have a direct formula here
 
 
 
 
we will use this formula here
so observe carefully
assume the first surface as 'f'
 
assume the second surface as 'g'
 
 
now we will calculate the gradient
 
 
to calculate this, we will differentiate the function with
respect to x * i
 
 
 
 
 
 
 
 
 
 
 
 
 
 
solve as follows
 
 
now keep the points
 
 
 
 
keep the formula of the angle between two surface
 
 
 
 
 
solve as follows
 
 
 
 
 
 
 
 
 
 
so this is the angle between two surfaces
now look at one more question
here the angle is given
let me explain it here
these are the two surfaces, there is an angle between them
 
 
 
they are making the angle at
 
 
 
so you have to calculate the value of A and B
the two surfaces are making the angle at this point
 
so assume one surface as 'f'
 
assume the second surface as 'g'
 
 
 
 
let me tell you one thing
this point is satisfying both 'f' and 'g'
because the angle is on that point
 
substituting the point
 
we will get
 
 
 
we have got the value for B
substitute the value of B
you will get the 'g' as
 
 
so this is our 'g'
and this we have 'f'
now we will calculate its angle
let me tell you
this is the formula for our angle
it is orthogonal so cos is 90 degree
so I will write here
 
let us calculate the gradient here
 
solve as before
 
 
 
 
now we will substitute the point
we will get
 
 
 
 
 
 
solve as follows
 
 
 
 
 
 
 
differentiating it with respect to x,y and z one by one
 
 
now we will put the points here
 
 
 
solve as follows
 
 
simplifying it
 
 
now we are given here the angle is 90 degree
and cos 90 is
it is 0 and when we will multiply this
solving it here
 
 
 
 
 
 
 
 
 
 
 
 
we will simplify it
 
 
 
so we will get the value as
 
 
 
we were asked in the question to find the value of a and b
 
so this is the way we solve such questions
in today's lecture we discussed
what is gradient
and solved questions based on it
how to calculate normal vector, unit normal vector
then we discussed about directional derivative
then we discussed how to calculate angle between two surfaces
so students keep watching my videos
and I think you are understanding it well
and if you are liking it don't forget to share my videos
 
and drop a comment, so we can come up with more good videos
 
 
