WEBVTT  generated by VCS
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Good day, ladies and gentlemen.
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I would like to extend a warm welcome to you
foranother contribution in the series
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electromagnetic theorie.
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Today it will be about the prior knowledge.
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We just want to take a look at what particularly
mathematical
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prior knowledge is necessary.
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The whole thing won't be a revision course,
but some things may
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when you see it, come back to me.
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My name is Hans Georg Krauthäuser, holder
of the professorship for
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electromagnetic theorie andcompatibility at
the TU Dresden.
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Good. Now we would start the presentation straight
away.
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Electromagnetic theory as we define it is
about the Maxwell equations.
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And the Maxwell equations, I have it for you
here
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shown again. There are 4 equations.
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We have the law of induction.
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We have expanded the Ampère law to include
the term
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which Maxwell brought about, the displacement
current density.
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We have the law that the magnetic field is
free of sources and we
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have Coulomb's law, which is where we introduced
the sources of the electric field.
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If we look at these basic equations, it becomes
obvious
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that in some areas knowledge is necessary
and quite obvious
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we need to know what fields are.
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We have to be able to handle the fields, simply
because the essential quantities,
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electrical and magnetic quantities that we
are considering here are field quantities.
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When we deal with fields, we should deal with
scalar as well as with
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deal with vector valued fields and also with
real and complex fields.
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Then these are partial differential equations,
that is, too
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immediately obvious that you are using the
differential operators on the fields
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have to be able to handle.
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The rotation and the divergence, but we will
then see that relatively quickly
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the gradient also plays a role, so that we
then already have the three essentials
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Differential operations here.
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There are partial differential equations and
if the basic equation is partial
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Are differential equations, then it is quite
obvious that we have a certain
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Must have knowledge of differential equations
and the reference to solution methods.
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But we will actually do a lot of that again
in the course of the course
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consider in detail.
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The formulation as it is given here above
, that is one
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differential wording, that is, which indicates
to you in a specific place
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instantaneously at this time, what are the
relationships and connections to one another?
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But it is often the case that we want to move
on to integral descriptions
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and must and in this transition from a differential
and local to
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integral description, since we will of course
have to fall back on
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the integral calculus, be it line, area or
volume integrals.
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We will do this in the coordinate system adapted
to the problem in each case
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make, so that the knowledge not only about
the Cartesian coordinate systems,
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but also via cylindrical coordinates and spherical
coordinates in 3D
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will definitely be necessary.
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In between, you also have to convert, this
is where the metric comes into play
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or the metric tensor and of course we use
integral theorems,
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especially the integral theorem of Gauss and
the integral theorem of Stokes, Kronecker
Delta
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will play a role, especially the delta function.
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I call this function here, as many of you will
know, it is strict
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not a function, but a disruption, but with
what we are here
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make, this distinction does not really matter.
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Let's go straight to the differential operators
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and we start here with the divergence.
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Divergence is a differential operation that
is used
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on a vector field and what comes out of it
is something that we also like
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Call source density and the source density
is a scalar field.
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The divergence of a vector field results in
a scalar field.
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You probably remember how to figure this out
in Cartesian coordinates.
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We use the div operator very often for divergence
, you will find that
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also written in the Nabla operator, then in
the form of a scalar product, i.e. Nabla
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scalar multiplied by the field F.
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And how do we calculate that in Cartesian coordinates?
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We calculate over this sum where d / d x_i
is the partial derivative of the ith
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Coordinate is and that is simply the shape
as we do one
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Would write scalar product.
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That is why it is very obvious to work with
this Nabla operator here.
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You will often find this abbreviated, and every
now and then we will too
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use this notation and, via Einstein's summation
convention, that
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just write as d_i F_i and the d_i is also
abbreviated here,
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the partial derivative with respect to the
ith component.
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And these are Cartesian coordinates.
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We don't list that here for all possible coordinate
systems  that can
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You can also look it up in the formulas.
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It makes sense, however, if you remember that
we
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That can also be written down without coordinates
and that results in coordinatesfree
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just as a limit over the volume V towards 0
and then here is essentially that
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Surface integral of the field over the closed
surface of this volume.
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n is the normal vector of the surface of the
volume.
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By convention, always directed outwards.
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And the whole thing is normalized with the
factor 1 by V.
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The rotation that we next envision will, in
turn, also
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applied to a vector field, but now also gives
a vector field as a result.
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And the rotation of the vector field is often
called the vortex strength of this
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Designated field. Here, too, first of all
in Cartesian coordinates
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Whole calculated or shown.
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red F is the usual abbreviation we will choose.
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Here, too, you will sometimes use Nabla XF.
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X is actually to be understood here in the
sense of a cross product if Nabla
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is the same as the vector (d / dx, d / dy,
d / dz), then just do the math here
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really by the rules of the cross point and
that's exactly what back here
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is written out again.
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Now, if you go through these sums one by one
, you get exactly that
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Expression that you would also get according
to the cross product.
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As a reminder: this epsilon i, j, k that appears
here is total
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antisymmetric unit tensor of the third order
or even that
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Levi called Cevita symbol. Here, too, we can
again use a compact form
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write down if we use Einstein's summation
convention again and
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We would simply have to write the partial derivative
a little more compactly
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this more compact form epsilon i, j, k unit
vector in idirection partial
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Derivation with respect to J and then just
applied to the kth field component.
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Again, very briefly: LeviCevita or totally
antisymmetric standard sensor
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third level: very simply defined in principle,
that is a matrix or a
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three dimensional entity  a tensor  which
simply has the value 1 if i, j, k
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are cyclic, for example 1, 2, 3; the value
is 1 if i, j, k are countercyclical
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are, for example 3, 2, 1 and in all other
cases is the value of
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Epsilon i, j, k equals 0.
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Here, too, there is a coordinatefree definition
as Limes over a
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Volume towards 0 and the decisive integral,
which is now taken here,
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so if n XF, n is again the normal vector at
the surface, that
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Integral again extends over the closed surface
of this volume and
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Here, too, normalization takes place with
the volume.
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These are very important operators that we
will need again and again here and
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yes, it can be stated that many also have
certain problems with it
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have and ask yourself “what is that?”.
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Perception in particular is sometimes the
problem.
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You can then work it out, but the notion is
not quite there.
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You have to say, there are some excellent
videos by Grant Sanderson
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whose channel I have now linked down here
and there are in particular
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a video about divergence and rotation, which
we will have a quick look at in a moment.
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The whole thing is not prepared from the formal
side , but comes
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actually in terms of the visual component.
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And Grant Sanderson uses an animation package
here  manim en Python
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Package that I have also linked again because
it is actually very much
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beautiful and also very beautiful to use.
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So let's take a quick look, here really in
the video about "Divergence and Rotation"
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so that you can get an impression.
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I took the sound down a little bit on the
video, you can watch it
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Watch the video at full volume on YouTube.
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But the bottom line is what we're seeing here
is really, really good
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Animation and that is what is often actually
missing, if you only think of it
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formal page.
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Take a look at the other videos on Grant Sanderson
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has on his channel.
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I am really impressed and consider this to
be an extremely good addition
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what we're doing here.
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Well, that should be enough here, turn it off
again and continue on
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the next slide and become the third very important
differential operator
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introduce  the gradient.
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The gradient does not appear immediately in
Maxwell's equations , but very quickly when
we introduce the potentials
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 at the latest when we use the gradient .
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The gradient is applied to a scalar field
and then turns it into a
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Vector field  the output is then a vector
field.
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And the gradient field is the directional field
of the strongest rise.
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And that is  I think  something that you
can imagine relatively well.
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So maybe we don't really need a video here.
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But I'll give you a source in a moment where
you can actually find it
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be able to look again through additional material.
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First in Cartesian coordinates: we will often
actually use the
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used full abbreviations, gradient of F, we
can do that too
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write as Nabla F, i.e. Nabla applied to F.
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By this we simply mean that we are following
the function  the field
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derive each other according to the components
 partially  and then just these contributions
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again weighted with the unit vectors in the
direction to sum up so that we
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so get a vector field in total.
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This would then be abbreviated again to the
Einstein sums convention
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simply e_i d_i F. It should also be mentioned
here that, of course, again a coordinatefree
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Representation gives, again over the limit
of a volume towards 0, also again
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with a standardization to volume 1 by V and
then that's just that
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Surface integral over the surface of the volume
F times the normal vector dS, that is
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relatively simply defined.
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I have already said that here, too, I think
there are actually good sources.
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I am referring to the "Khan Academy"  web
address is given here.
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Here, too, you can find a video, directly on
the gradient, but it is worthwhile with the
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"Khan Academy" maybe to look at other things.
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In contrast to YouTube, not only videos are
shown here, but there is also
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many topics actually also real courses with
exercises, with selftest exercises,
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where you can click through, certainly a good
addition for many things
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to quickly work on something that is no longer
so present.
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Well, those were the differential operators
that we will come back to over and over again.
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I already said at the beginning that of course
we also need the whole thing in
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different coordinate systems.
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This is simply because the Cartesian coordinates
are not always used
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are problemadapted coordinates and problems
are easiest to solve when
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when you have found a good coordinate system
that is adapted to the problem .
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Often you come here in addition to the Cartesian
coordinates that you all know,
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with cylindrical or spherical coordinates
.
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The whole thing in 2D or in 3D, of course.
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We typically don't need the higher dimensions.
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You need to know how to convert the components
.
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This then leads you very quickly to the concept
of the Jacobi matrix, which you then have
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also need to convert differential or integral
elements.
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That means around div, red, degree in different
coordinate systems
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to represent, but also the volume elements,
the surface elements or
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to display the line elements in different coordinates.
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The metric tensor, the metric, then plays
a very decisive role
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Role that you have just calculated from the
Jacobi matrix or with the help of the Jacobi
matrix
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calculate. Integral sentences is something
that you actually need all the time.
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If you have the derivations and take the appropriate
forming steps, need
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one is actually constantly in particular the
Gauss theorem and the Stokes theorem.
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Let's start with Gauss's theorem.
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Gauss's theorem states that the surface integral
of a vector field F
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over a closed area O of V, surface of V, is
equal to that
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The volume integral of the divergence of F
extends over that of the area O of V
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including volume or, formally expressed mathematically:
volume integral
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Divergence F dV is equal to the surface integral
F times n dS.
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Here again n is the normal vector.
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Of course, this also works in two dimensions,
we wrote down
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this now for a threedimensional volume and
surface.
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These are terms that are used all over again
in three dimensions,
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But formally it is also possible in twodimensional,
then the volume
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a surface and the surface integral would then
be flat
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the closed path integral.
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Stokes' theorem is similarly simple and you
should be familiar with it, yes, how
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said in the sense of repeating a brief invocation
of the sentences is here
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maybe just fine if we go over it again .
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The curve or line integral of a vector field
F along a simple
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closed curve C of A, contour of A, is equal
to the surface integral of
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Rotation of F over any surface A bounded by
curve C.
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Or again expressed somewhat formally: the
path integral F ds around the contour of the
surface
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A is equal to the area integral rotation F
dS.
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And of course we can write that again as integral
rotation F scalar
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multiplied by the normal vector n dS.
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Something that may have been a little more
forgotten  something for you
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but the delta function is not new either.
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Or rather, it is actually a delta distribution,
because there is none
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closed function expression for what we're
about to introduce here.
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But we'll see that in a moment.
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But let's get down to motivation first.
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Why do we actually need the delta function,
the delta distribution?
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00:22:20.020 > 00:22:23.020
What do we do with it? Why is it so important?
225
00:22:23.020 > 00:22:36.020
Here is an example: We all know the Coulomb
field, the one for a point charge
226
00:22:36.020 > 00:22:41.020
looks exactly like I have shown you here.
227
00:22:41.020 > 00:22:46.020
That is, the field E is equal to a normalization
 1 by 4 pi eplsilon_0 
228
00:22:46.020 > 00:22:52.020
Q is the charge at the origin and then the
1 drops off through r ^ 2 and is always in
229
00:22:52.020 > 00:22:56.020
Direction of the unit vector directed in the
radial direction.
230
00:22:56.020 > 00:23:04.080
In other words, a simple size that you definitely
already know from the basics of electrical
engineering.
231
00:23:04.080 > 00:23:10.080
If we now consider the divergence  the source
strength  of such a point charge
232
00:23:10.080 > 00:23:19.080
want to calculate, then we expect that those
outside the origin, there where
233
00:23:19.080 > 00:23:26.080
are no sources, is equal to 0 and at the origin,
where the source is, there should be
234
00:23:26.080 > 00:23:33.080
the source strength actually yield a value,
yield a finite value.
235
00:23:33.080 > 00:23:38.080
And we have just said: the divergence is what
the source strength calculates.
236
00:23:38.080 > 00:23:42.080
And if we now have all the constants that
we are up here
237
00:23:42.080 > 00:23:46.080
have to leave out, then the tasks are limited
to us doing the
238
00:23:46.080 > 00:23:52.080
Calculate divergence from 1 through r ^ 2 unit
vector in direction R.
239
00:23:52.080 > 00:23:58.080
The problem is spherically symmetric.
240
00:23:58.080 > 00:24:05.040
So we would certainly go there and use the
divergence operator directly
241
00:24:05.040 > 00:24:08.040
apply in spherical coordinates.
242
00:24:08.040 > 00:24:10.040
We look for that from the formulas,
243
00:24:10.040 > 00:24:11.040
when we don't know anymore
244
00:24:11.040 > 00:24:17.040
It is clear that derivatives according to
d / d phi and d / d theta will of course also
appear,
245
00:24:17.040 > 00:24:21.040
But they don't play a role here because we
don't have any here
246
00:24:21.040 > 00:24:24.040
Have dependence on phi and theta.
247
00:24:24.040 > 00:24:35.040
That means, the only thing left here is the
derivative in the radial direction
248
00:24:35.040 > 00:24:39.040
dr and if we look at the formulary and how
that
249
00:24:39.040 > 00:24:44.040
looks, then it looks like it is written here:
1 through r ^ 2
250
00:24:44.040 > 00:24:53.040
Derivation in r direction and then r ^ 2 times
the function we want to derive and
251
00:24:53.040 > 00:24:56.040
that's 1 through r ^ 2 e_r.
252
00:24:56.040 > 00:25:06.000
It doesn't look that complicated at first
, but at this point it has to
253
00:25:06.000 > 00:25:15.000
which we want to consider, namely at the origin,
exactly one small problem: r ^ 2
254
00:25:15.000 > 00:25:20.000
by r ^ 2 at this point we can of course shorten
it out  that's just 1
255
00:25:20.000 > 00:25:28.000
 so that what remains: the partial derivative
in the radial direction of the
256
00:25:28.000 > 00:25:31.000
Unit vector in the radial direction.
257
00:25:31.000 > 00:25:37.000
Outside the origin there is no problem at
all : you have the unit vector,
258
00:25:37.000 > 00:25:47.000
the points in the radial direction and if you
change in the same direction
259
00:25:47.000 > 00:25:50.000
look at yourself, you will find that the unit
vector is in that direction
260
00:25:50.000 > 00:25:52.000
not changed at all.
261
00:25:52.000 > 00:25:59.000
That means, outside of the origin it is simply
0, which means no change
262
00:25:59.000 > 00:26:08.060
Derivation is 0 and that's exactly what we
actually want here.
263
00:26:08.060 > 00:26:27.060
But if we look at the origin and at the origin
you must now
264
00:26:27.060 > 00:26:32.060
imagine what does the unit vector look like
at the origin?
265
00:26:32.060 > 00:26:38.060
If you have the origin at this point, then
the unit vector will be yes 
266
00:26:38.060 > 00:26:45.060
if you imagine infinitisimally you are moving
away from the origin  then
267
00:26:45.060 > 00:26:52.060
he points sometimes in this direction and sometimes
in this direction and that is exactly that
268
00:26:52.060 > 00:26:55.060
Dilemma what you have at this point.
269
00:26:55.060 > 00:27:02.020
If you really want to calculate the derivative
now , then that is in principle
270
00:27:02.020 > 00:27:03.020
really indefinite.
271
00:27:03.020 > 00:27:09.020
You don't know exactly what the unit vector
looks like, even with
272
00:27:09.020 > 00:27:12.020
an infinitimal shift out of the origin.
273
00:27:12.020 > 00:27:18.020
In other words, you actually have a problem
at this point  for the origin.
274
00:27:18.020 > 00:27:26.020
And because of these kinds of problems, not
just like now with the Coulomb potential
275
00:27:26.020 > 00:27:30.020
appear, but that appear again and again with
similar things.
276
00:27:30.020 > 00:27:36.020
It is precisely because of such problems that
we actually run the delta function  the
277
00:27:36.020 > 00:27:38.020
Delta distribution  a.
278
00:27:38.020 > 00:27:46.020
We now simply define it once by the properties
that it has.
279
00:27:46.020 > 00:27:51.020
And that's two pieces: We just say that there
should be a function  one
280
00:27:51.020 > 00:27:56.020
Disputation  for which it should apply that
it vanishes for x not equal to 0,
281
00:27:56.020 > 00:28:13.080
so delta (x) = 0 for x not equal to 0 and that
if I take the integral of this function
282
00:28:13.080 > 00:28:18.080
from infinite to + infinite, that it should
then be normalized.
283
00:28:18.080 > 00:28:20.080
That is, this integral is simply 1.
284
00:28:20.080 > 00:28:25.080
The area under the curve infinity to + infinity
is 1.
285
00:28:25.080 > 00:28:35.080
Such functions actually only exist in the
limit value of other functions.
286
00:28:35.080 > 00:28:38.080
That is exactly why it is not a function itself.
287
00:28:38.080 > 00:28:44.080
But we can actually only define that as the
limit of n versus
288
00:28:44.080 > 00:28:56.080
infinite of functions f_n of x, which are
just normalized and for which then just
289
00:28:56.080 > 00:29:05.040
simply this value f_n (x) for n towards infinity
towards 0 strives outside of the
290
00:29:05.040 > 00:29:13.040
Origin. There are such series of functions.
291
00:29:13.040 > 00:29:17.040
You can, for example, define it in sections
, as you did here .
292
00:29:17.040 > 00:29:22.040
You can easily convince yourself that this
suite of functions does just that
293
00:29:22.040 > 00:29:26.040
Property that I have requested above.
294
00:29:26.040 > 00:29:31.040
But that doesn't have to be a split definition,
we can do it
295
00:29:31.040 > 00:29:40.040
make here with such a Gaussian function or
through such a sine x
296
00:29:40.040 > 00:29:44.040
x ^ 2 row that works too.
297
00:29:44.040 > 00:29:54.040
These are three examples of how you can add
a series of functions to then
298
00:29:54.040 > 00:30:00.000
to come to the delta function even in the limit
value towards infinity .
299
00:30:00.000 > 00:30:07.000
There are important relationships for the
delta function, which we will also briefly
describe here
300
00:30:07.000 > 00:30:10.000
want to perform, which we want to repeat very
briefly .
301
00:30:10.000 > 00:30:20.000
From the definition it follows quite directly
an important property.
302
00:30:20.000 > 00:30:29.000
If you integrate via a function and then also
as a factor in the integral
303
00:30:29.000 > 00:30:37.000
have the delta function, then there simply
comes the function value at the origin
304
00:30:37.000 > 00:30:39.000
if X equals 0 out.
305
00:30:39.000 > 00:30:44.000
You can do this via the definition of the delta
function and via the
306
00:30:44.000 > 00:30:48.000
Easily check the standardization property.
307
00:30:48.000 > 00:30:53.000
For x equal to 0, this is just multiplying
a 0 , that's why they play
308
00:30:53.000 > 00:30:58.000
other function values are irrelevant
and the normalization then means that
309
00:30:58.000 > 00:31:00.060
which is simply f (0).
310
00:31:00.060 > 00:31:06.060
This is a very, very essential property that
you actually also have
311
00:31:06.060 > 00:31:10.060
can use if you move the whole thing now .
312
00:31:10.060 > 00:31:16.060
That is, if you do not look at delta of x
now , but if you look at it
313
00:31:16.060 > 00:31:21.060
Delta of x minus a, then you can easily reconsider
that
314
00:31:21.060 > 00:31:29.060
all values, except x equals a, do not play
a role here , because then delta
315
00:31:29.060 > 00:31:31.060
of x minus a equals 0.
316
00:31:31.060 > 00:31:37.060
And of course the normalization remains the
same, even if you do that on the axis
317
00:31:37.060 > 00:31:44.060
shift something so that here f of a simply
comes out.
318
00:31:44.060 > 00:31:50.060
One is also important, especially for physical
problems
319
00:31:50.060 > 00:31:58.060
Scaling property. You can imagine that what
we are looking at
320
00:31:58.060 > 00:32:03.020
does not always have to be a dimensionless
quantity , but sometimes it is
321
00:32:03.020 > 00:32:07.020
dimensional quantity  could be a length ,
for example .
322
00:32:07.020 > 00:32:15.020
And then it's really good to know how the
delta function behaves when
323
00:32:15.020 > 00:32:19.020
one just multiplies here with a scalar k.
324
00:32:19.020 > 00:32:22.020
The k could now be the unit, for example .
325
00:32:22.020 > 00:32:26.020
And then this relationship, which I have written
down for you here, applies 1 through k
326
00:32:26.020 > 00:32:28.020
Amount delta of x.
327
00:32:28.020 > 00:32:39.020
Of course, we can also generalize the delta
function to threedimensional conditions:
328
00:32:39.020 > 00:32:42.020
It's written down here.
329
00:32:42.020 > 00:32:53.020
So if r is x e_z + y e_y + z e_z , then we
can define delta of r
330
00:32:53.020 > 00:32:59.020
as the product of three delta functions, delta
of x times delta of y times delta of z.
331
00:32:59.020 > 00:33:09.080
And then the same definition applies again:
if one of the quantities x, y or
332
00:33:09.080 > 00:33:21.080
z is not equal to zero, then this total product
results in 0 and the normalization applies
333
00:33:21.080 > 00:33:25.080
of course again accordingly  you can easily
think about that  so that
334
00:33:25.080 > 00:33:30.080
we also use this delta distribution in three
dimensions or in any
335
00:33:30.080 > 00:33:33.080
Can use dimensions.
336
00:33:33.080 > 00:33:43.080
Yes, now we can actually do some math and
the Gaussian integral theorem
337
00:33:43.080 > 00:33:52.080
and use the delta function to actually get
this problem  what's up
338
00:33:52.080 > 00:33:58.080
the divergence 1 through r ^ 2 e_r  to approximate
again.
339
00:33:58.080 > 00:34:06.040
And to do that, let's just build a volume
integral
340
00:34:06.040 > 00:34:09.040
over divergence 1 through r ^ 2 e_r.
341
00:34:09.040 > 00:34:13.040
And we are completely free as to which surface
we use.
342
00:34:13.040 > 00:34:18.040
And it is obvious that when we have such a
problem, yes
343
00:34:18.040 > 00:34:24.040
Obviously a spherical problem is that we then
also have a special volume
344
00:34:24.040 > 00:34:28.040
because we can make our lives a little easier
with it.
345
00:34:28.040 > 00:34:35.040
The volume we choose here is simply a sphere
of radius R so K (R)
346
00:34:35.040 > 00:34:42.040
should now simply be a sphere around the origin
with radius R.
347
00:34:42.040 > 00:34:51.040
Then Gauss's theorem tells us that we can
convert this volume integral into
348
00:34:51.040 > 00:35:00.000
a surface integral over the surface of the
sphere and then the
349
00:35:00.000 > 00:35:02.000
Divergence simply left out.
350
00:35:02.000 > 00:35:09.000
And here at the back there is of course the
surface element.
351
00:35:09.000 > 00:35:17.000
And this surface integral extends over the
surface of the sphere, that
352
00:35:17.000 > 00:35:23.000
obviously only has contributions at r = R.
353
00:35:23.000 > 00:35:27.000
We can also see why it was so cheap to take
a bullet.
354
00:35:27.000 > 00:35:30.000
That makes the whole thing a lot easier.
355
00:35:30.000 > 00:35:40.000
And of course we can also write this surface
element as r ^ 2, sin
356
00:35:40.000 > 00:35:43.000
theta, d theta d phi.
357
00:35:43.000 > 00:35:49.000
And it is always directed outwards, so e_r.
358
00:35:49.000 > 00:36:00.060
This is quite simply the surface element of
this sphere with radius R around the origin.
359
00:36:00.060 > 00:36:06.060
We can use that, it makes the whole thing
pretty easy for us now: We then have
360
00:36:06.060 > 00:36:14.060
obviously an integration over d phi  it runs
from 0 to 2 pi.
361
00:36:14.060 > 00:36:19.060
The expression that we have under the integral
is not phi at all
362
00:36:19.060 > 00:36:27.060
dependent, so that this integral d phi simply
gives us a factor of 2 pi.
363
00:36:27.060 > 00:36:36.060
That goes from 0 to 2 pi and the integral d
phi from 0 to 2 pi is just 2 pi and es
364
00:36:36.060 > 00:36:41.060
Then of course the integration d theta remains
 it extends over the range from 0 to pi.
365
00:36:41.060 > 00:36:48.060
And otherwise, the area integral is simply
inserted here and we still have
366
00:36:48.060 > 00:37:00.020
immediately calculated that e_r times e_r is
simply equal to 1 and that here in this one
367
00:37:00.020 > 00:37:03.020
Expression no longer appears at all.
368
00:37:03.020 > 00:37:10.020
Yes, and there is again an r ^ 2 through r
^ 2  that's a 1, so that in the end
369
00:37:10.020 > 00:37:13.020
all that remains is the integral over sin theta
d theta.
370
00:37:13.020 > 00:37:21.020
And that's not difficult either: the integral
0 to pi sin theta d theta is 2.
371
00:37:21.020 > 00:37:28.020
And that with the 2 pi that we already have,
simply gives a factor of 4 pi.
372
00:37:28.020 > 00:37:41.020
That tempts us now, but simply the expression
we are here for the
373
00:37:41.020 > 00:37:50.020
Divergence have to be identified with 4 pi
delta of r.
374
00:37:50.020 > 00:37:57.020
Because if we do that, imagine we'd get 4
pi delta of r up here
375
00:37:57.020 > 00:38:02.080
insert, then the 4 pi goes forward, of course.
376
00:38:02.080 > 00:38:13.080
And the delta of r over this sphere gives
us that outside of the origin
377
00:38:13.080 > 00:38:19.080
no more contributions at all and is standardized
overall.
378
00:38:19.080 > 00:38:24.080
Of course we can then let the ball go to infinity
, which does not change the value,
379
00:38:24.080 > 00:38:31.080
because what is under the integral has no
more contributions at all, not equal to 0.
380
00:38:31.080 > 00:38:37.080
And then 4 pi is equal to 4 pi on both sides,
which is obviously correct.
381
00:38:37.080 > 00:38:44.080
And with that we would have shown that this
identity applies.
382
00:38:44.080 > 00:38:53.080
In other words, we have succeeded in calculating
this divergence here , in particular
383
00:38:53.080 > 00:38:55.080
also to be calculated for the origin.
384
00:38:55.080 > 00:39:01.040
And that gives a value that is linked to the
delta function.
385
00:39:01.040 > 00:39:09.040
But that's not all that surprising and ultimately
comes from the fact that
386
00:39:09.040 > 00:39:15.040
Construct what we originally chose, namely
to say we have here
387
00:39:15.040 > 00:39:21.040
a point charge in the origin, that's something
very, very artificial.
388
00:39:21.040 > 00:39:28.040
In fact, there is no such thing as a point
charge; every charge will have a certain extent.
389
00:39:28.040 > 00:39:33.040
And if one were to count on a finite extent,
then we would have
390
00:39:33.040 > 00:39:36.040
in fact not this problem at the origin.
391
00:39:36.040 > 00:39:42.040
That means that we get the delta function into
the expression for the
392
00:39:42.040 > 00:39:46.040
Divergence on the one hand is quite nice when
we link it in this way
393
00:39:46.040 > 00:39:49.040
the divergence and the delta function at the
point.
394
00:39:49.040 > 00:39:56.040
And on the other hand we don't need to be confused
395
00:39:56.040 > 00:40:03.000
because what we have started is also something
very  yes, artificial wants
396
00:40:03.000 > 00:40:06.000
I might not even say  model.
397
00:40:06.000 > 00:40:12.000
And that a strange solution then comes out
for the source strength on
398
00:40:12.000 > 00:40:17.000
The origin is not really surprising.
399
00:40:17.000 > 00:40:27.000
Of course, when we have already reached this
point, we can
400
00:40:27.000 > 00:40:30.000
just keep calculating.
401
00:40:30.000 > 00:40:37.000
And something that can be calculated very easily
is actually the gradient 1 / r.
402
00:40:37.000 > 00:40:41.000
Just write down and it will be clear relatively
quickly.
403
00:40:41.000 > 00:40:46.000
Of course there is no phi and no theta component
here, so this is easy
404
00:40:46.000 > 00:40:50.000
stops: e_r d after dr from 1 / r.
405
00:40:50.000 > 00:40:56.000
And that's quite obviously 1 / r ^ 2 e_r.
406
00:40:56.000 > 00:41:04.060
And that is, apart from the sign, exactly the
term we are talking about here under the divergence
407
00:41:04.060 > 00:41:10.060
 of which we already know how great the divergence
is.
408
00:41:10.060 > 00:41:21.060
And if we bring that together and now just
use this expression up here
409
00:41:21.060 > 00:41:28.060
we get an important relationship for the Laplace
operator we would
410
00:41:28.060 > 00:41:30.060
also introduce on this occasion.
411
00:41:30.060 > 00:41:38.060
The Laplace operator is nothing more than
divergence applied to the
412
00:41:38.060 > 00:41:46.060
Gradient field. It is also very often written
with such a delta, with
413
00:41:46.060 > 00:41:53.060
the Laplace operator. And then immediately
after the insertion we see : Laplace 1 / r

414
00:41:53.060 > 00:42:01.020
which is nothing else than divergence gradient
of 1 / r  and that is equal to 4 pi delta
of (r).
415
00:42:01.020 > 00:42:09.020
The minus sign is exactly the minus that is
already here , so don't be surprised
416
00:42:09.020 > 00:42:15.020
if you insert it into the divergence term
or the argument of the
417
00:42:15.020 > 00:42:20.020
Divergence, then the minus sign appears on
the other
418
00:42:20.020 > 00:42:23.020
Side up, no problem at all.
419
00:42:23.020 > 00:42:35.020
Yes, and now we're almost at the end of the
inventory for that
420
00:42:35.020 > 00:42:37.020
Prior knowledge that we need here.
421
00:42:37.020 > 00:42:45.020
And I don't want to end this without actually
going back to it
422
00:42:45.020 > 00:42:49.020
the decomposition of vector fields to come.
423
00:42:49.020 > 00:42:55.020
Namely, on the socalled Helmholtz theorem
 often that also means
424
00:42:55.020 > 00:43:05.080
the fundamental theorem of vector analysis
and is really extremely important and delivers
425
00:43:05.080 > 00:43:11.080
us mathematical background for much of what
we do.
426
00:43:11.080 > 00:43:18.080
First of all, the Helmholtz theorem provides
a very simple statement.
427
00:43:18.080 > 00:43:24.080
This is the case when we have a fairly arbitrary
vector field f.
428
00:43:24.080 > 00:43:31.080
And quite arbitrarily: we want to assume here
that it is actually stronger than 1 / r
429
00:43:31.080 > 00:43:36.080
towards infinity drops to 0.
430
00:43:36.080 > 00:43:39.080
We'll see later what we actually need it for.
431
00:43:39.080 > 00:43:45.080
But that is a condition for actually relevant
problems
432
00:43:45.080 > 00:43:48.080
always fulfilled  so no strict restriction.
433
00:43:48.080 > 00:43:52.080
And we can assume that practically every vector
field we use in
434
00:43:52.080 > 00:43:58.080
relevant problems are encountered, precisely
this property is fulfilled.
435
00:43:58.080 > 00:44:07.040
So, a vector field that drops sufficiently
for r towards infinity can be
436
00:44:07.040 > 00:44:18.040
decompose into a rotationfree part a and
a divergencefree part b.
437
00:44:18.040 > 00:44:28.040
That is, we can write F (r) as a (r) plus
b (r), being the rotation of
438
00:44:28.040 > 00:44:34.040
a disappears and the divergence of b disappears.
439
00:44:38.040 > 00:44:50.040
Furthermore, we note that gradient fields
are rotationfree.
440
00:44:50.040 > 00:44:58.040
This means that the rotation gradient phi always
equals 0 for any scalar field phi.
441
00:44:58.040 > 00:45:05.000
And we note that fields of rotation are free
of divergence.
442
00:45:05.000 > 00:45:11.000
This means that the divergence of rotation
A applies to any vector field A.
443
00:45:11.000 > 00:45:13.000
is also 0.
444
00:45:13.000 > 00:45:22.000
This is ultimately the construction manual
for how we now fields a and b
445
00:45:22.000 > 00:45:29.000
so that rotation of a vanishes and divergence
b vanishes.
446
00:45:29.000 > 00:45:35.000
Yes, it will now be done in such a way that
a just a
447
00:45:35.000 > 00:45:41.000
Gradient field and b is just a field of rotation.
448
00:45:41.000 > 00:45:44.000
We can write that down.
449
00:45:44.000 > 00:45:51.000
This is followed by the representation with
a scalar potential phi and a
450
00:45:51.000 > 00:46:00.060
Vector potential A that F (r)  any vector
field that is just sufficient
451
00:46:00.060 > 00:46:07.060
drops sharply towards infinity  can be written
as  gradient phi + rotation
452
00:46:07.060 > 00:46:13.060
So A  gradient of a scalar potential + rotation
of a vector potential.
453
00:46:13.060 > 00:46:19.060
And at the moment you might be wondering about
the minus that appears here.
454
00:46:19.060 > 00:46:24.060
This is pure convention  it really doesn't
matter.
455
00:46:24.060 > 00:46:30.060
We just write it down so that there is no
confusion later .
456
00:46:30.060 > 00:46:37.060
You could just as easily pull the minus sign
into the potential  then everything would
457
00:46:37.060 > 00:46:40.060
others apply the same way  it is a pure conversion.
458
00:46:40.060 > 00:46:53.060
The whole thing wouldn't be so nice if we
didn't actually do this
459
00:46:53.060 > 00:46:58.060
Could specify scalar potential and the vector
potential.
460
00:46:58.060 > 00:47:00.020
But that is exactly the case.
461
00:47:00.020 > 00:47:08.020
That means, you can actually now both the
scalar  here is a
462
00:47:08.020 > 00:47:13.020
Spelling mistakes  I'll correct that at some
point  of course with a scalar
463
00:47:13.020 > 00:47:20.020
k still in it  this scalar potential and the
vector potential can now
464
00:47:20.020 > 00:47:27.020
can actually be calculated directly from the
field F in the manner
465
00:47:27.020 > 00:47:33.020
as it stands here: that means, these are volume
integrals once over the divergence
466
00:47:33.020 > 00:47:36.020
and once about the rotation of the field.
467
00:47:36.020 > 00:47:42.020
To make it really clear, I've also added lines
here:
468
00:47:42.020 > 00:47:47.020
these differential operators now refer to
the deleted coordinate
469
00:47:47.020 > 00:47:53.020
and not on the unprimed coordinate on the
outside.
470
00:47:53.020 > 00:47:55.020
Otherwise you could just drag that in front
of the integral.
471
00:47:55.020 > 00:48:01.080
It actually has to be derived here with regard
to the dashed coordinates.
472
00:48:01.080 > 00:48:13.080
That is the expression. So, in that form, it
actually applies to fields F that are fast
enough  that is
473
00:48:13.080 > 00:48:20.080
faster than 1 / r  strive towards 0.
474
00:48:20.080 > 00:48:27.080
You can expand this in the event that this
requirement is not met.
475
00:48:27.080 > 00:48:35.080
Then there are actually additional surface
terms, each with two
476
00:48:35.080 > 00:48:41.080
Surface integrals over the surface of a volume.
477
00:48:41.080 > 00:48:48.080
And yes, here you can already read a bit where
this condition comes from.
478
00:48:48.080 > 00:48:56.080
So if I want this thermal bath to be eliminated,
then I just have to see to it that
479
00:48:56.080 > 00:49:06.040
F (r ') tends towards 0 faster than 1 / rr'
tends towards finite.
480
00:49:06.040 > 00:49:16.040
And that is exactly this condition that r times
F for r vanishes towards infinity.
481
00:49:16.040 > 00:49:23.040
Then the integrand in this surface integral
drops off sufficiently quickly, see above
482
00:49:23.040 > 00:49:29.040
that even if I make the volume big enough
, the surface of the sphere won't go fast
483
00:49:29.040 > 00:49:34.040
grows enough to compensate for this drop in
F.
484
00:49:34.040 > 00:49:40.040
And in that case the surface terms don't matter,
and it does
485
00:49:40.040 > 00:49:44.040
what remains is exactly what I have just presented.
486
00:49:44.040 > 00:49:49.040
That is the term that we will use in the future
 as a rule  and only
487
00:49:49.040 > 00:49:53.040
in very, very big exceptional cases, if there
is no other way, then just come
488
00:49:53.040 > 00:49:57.040
this term, add these terms.
489
00:49:57.040 > 00:50:04.000
Often times they are actually called these
components as well
490
00:50:04.000 > 00:50:09.000
find transversal and longitudinal components
labeled.
491
00:50:09.000 > 00:50:17.000
So what results from the gradient field 
from the scalar potential  becomes
492
00:50:17.000 > 00:50:23.000
often referred to as the longitudinal component
and the
493
00:50:23.000 > 00:50:29.000
divergencefree component rotation A, which
therefore comes from the vector potential,
the
494
00:50:29.000 > 00:50:35.000
the transverse component of the field is also
very often referred to.
495
00:50:35.000 > 00:50:44.000
The background is in the place of that when
you do a Fourier transform
496
00:50:44.000 > 00:50:52.000
of the field F and then look at the whole thing
at a point k  then are
497
00:50:52.000 > 00:50:58.000
You in kspace  and then consider a point
k, which is of course a vector
498
00:50:58.000 > 00:51:06.060
you can split the field in kspace in a contribution
in the direction of
499
00:51:06.060 > 00:51:13.060
k and in a post perpendicular to k: i.e. once
longitudinally and once
500
00:51:13.060 > 00:51:19.060
transversal. And it turns out that these posts
are exactly those posts
501
00:51:19.060 > 00:51:25.060
that we have just mentioned here.
502
00:51:25.060 > 00:51:34.060
This means that rotation A is precisely this
transverse component and gradient phi
503
00:51:34.060 > 00:51:40.060
or  Gradient phi is then just the longitudinal
contribution.
504
00:51:40.060 > 00:51:52.060
Good. That should actually be enough for this
short introduction.
505
00:51:52.060 > 00:51:56.060
Thank you very much for your attention.
506
00:51:56.060 > 00:52:04.020
Of course it is the case that we always do
one or the other later
507
00:52:04.020 > 00:52:07.020
will repeat math.
508
00:52:07.020 > 00:52:13.020
But we will simply ignore some things and your
task
509
00:52:13.020 > 00:52:17.020
it will then be that in your documents that
you have from others
510
00:52:17.020 > 00:52:22.020
Lectures or a good book, then just look again.
511
00:52:22.020 > 00:52:26.020
Well that was it. As I said: Thank you very
much!
512
00:52:26.020 > 00:52:30.020
Please refer to the website for more information
if necessary.
513
00:52:30.020 > 00:52:32.020
Until next time.