Laplacian Operator Video Lecture | Electromagnetic Fields Theory (EMFT) - Electrical Engineering (EE)

okay so this year we're going to be
talking about another type of operator
that is commonly found in vector
calculus and it is called the laplacian
operator so basically means is that you
have this nabla operator or this del
operator and it looks like it's square
but it's not action square it just
simply means that you have the second
derivative of some scalar functions so
the way this works is as folks you're
going to apply this to some scalar
function Phi and basically you're just
going to add up the second derivatives
of that scalar function so you're going
to have the following five plus Delta
double zero Phi so those are the
secondary T's respect to each of those
variables now what does this actually
come from well the idea is that if you
have the gradient of a scalar field you
know that this is actually going to turn
out to be a vector so it's it can be
written in the following manner we have
partial of Y with respect to Phi and
then partial with respect to set of
value
now in order to obtain an expression for
the second derivative if we want to
return to some kind of scalar field from
here since this is a vector well we need
to take the dot product and that's going
to return the expression to a scalar
field so one thing we could do is we
could take the dot product with respect
to the del operator once again of this
gradient of of this gradient field which
is a vector e itself and then this is
going to be as false we know that the
dot operator is defined as the first
derivative and it is a vector so you
have this and then SEC and you're going
to take the dot product with respect to
this right here so we have this Phi or
Phi and then partial with respect to set
of Phi now if you multiply element by
element and then just add them together
this is going to give you this
operating on partial effects
like to Phi and then partial of Y
operating on partial of Y with respect
to Phi and then finally partial set
acting on the partial set of Phi and we
know from the properties of operators so
basically if you have the following
differential operator acting on another
differential operator what happens is
that this is going to turn out to be an
operator of higher order so basically
this is going to become a second
derivative that looks like this and
basically in shorthand notation which is
represented in this way so what's
happening here is you're multiplying
these operators together so this is
exactly what's going to happen so this
is going to become this expression right
here then this one is going to become
this expression here and finally you're
going to have this expression so it
turns out that by taking the dot product
of del operator with respect to the
gradient of a scalar function do you
obtain the laplacian of that or the
laplacian operator acting on that
expression and then basically this
returns us to the original problem now
where might this actually be useful well
the thing is that this is used quite a
lot in describing partial differential
equations so just to give you a few
examples of where you might see this
kind of rotation one of the main ones
you might come across at some point is
called the Laplace equation which is
essentially just the laplacian operator
acting on some scalar function so if you
were to write this in something like two
dimensions you would do the following
plus partial of Phi with respect to Y
squared and this is equal to zero so
this would be a second order linear
partial differential equations and you
notice that it is homogeneous so there's
no function on the other side is
basically just the derivatives another
another type of equation where you might
see this is called the Poisson equation
which is just like the Laplace equation
but it's just the inhomogeneous case so
you would
is equal to some other function and then
the idea is kind of the same and then
you will have equations in which you
will kind of just combine these things
together so you might have something
like this and then this might be equal
to some partial derivative of the
function with respect to time so in that
case what you would have is something
like the wave equation or something like
a heat equation so there are a lot of
things that that can be used and
normally because this is such a
convenient way of running it this is the
kind of rotation that is used so the
laplacian operator in itself it's not
something that we use to compute things
but it is something that we used to
represent things in a more compact form
and we'll just to show you how this can
be used like let's have a look at some
things we can do so let's say we have
the following scalar function so let's
have the function equal to x squared y
plus set cubed x plus e to the X Y
alright so now the first thing we need
to do is well we can just take the
gradient of this and then take the dot
product or take the divergence of that
gradient what we can do is just take the
look used to apply an operator to
operate on it directly so this is what's
going to happen you're going to have
this operate on the whole function plus
secondary t with respect to Y operating
the whole function and then secondary T
respective set acting on the whole
function again so in the first case if
you differentiate this twice with
respect to X for you came together the
first time around actually I'm just
going to do it yeah it's going to be a
little bit complicated but let's do it
so you're going to get to X Y so this is
the first derivative with respect to X
and then we're going to have plus set
cubed and then here we're going to have
plus y e to the X Y and of course this
is going to become because this is the
first derivative respect to X of the
whole scalar function we want the second
derivative so we differentiate the
again with respect to X we get two y
plus y squared e to the X Y and then
here what are we going to get well let's
differentiate with respect to Y once
first so this is going to become x
squared and then here we're going to
have plus X e to the X Y and then once
we differentiate it again with respect
to Y this becomes x squared e to the X Y
and then finally with respect to set
Bowl this is just going to be 3z squared
times X and then one more time this is
going to become 6 Z X so now if we put
it all this together it turns out that
the laplacian of this scalar field is
going to be 2 y plus y squared e to the
XY plus x squared e to the XY plus 6 set
X so that is it that is basically what
we're going to get we could have gotten
the same thing if we just compute the
gradient of this function first and then
just apply the divergence operation to
it so that's something that you can do
just to show that this operation here is
equivalent to that but this is the
general idea and the main the main thing
that we used to apply and which is the
represent partial differential equations
are very common in physics in a more
compact form so we won't do all too much
in this it is just to let you know that
this kind of operator exists it is just
called a flashing and it is just a
scalar sum of all the second derivatives
of a scalar function or a scalar field