James Clerk Maxwell
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- Gautam Sengupta and H C Verma.
Hello sir,
ReplyDeletethis is himanshu. i seriously fail to understand the grad
behaviour at maxima or Extrema...
This is not too difficult. Just as dy/dx gives slope for 1 variable the grad of phi gives the slope for 3 dimensions/variables. At maxima minima
ReplyDeleteor at a point of inflection dy/dx is zero similarlly
grad of phi is zero at a maxima, minima or inflection in 3 dimensions.
It is easier to visualise in 2 dimensions. Imagine you are on the surface of a hill.
At the top of the hill the slope is zero.
It is also zero in the bottom point of a valley.
Also zero at flat kinks on the surface like a
ledge on the hillside.
If you still do not understand please contact me after the lectures.
a laplacian operator can be used as a scalar in calculations. it does not need calculation in the same way as the del operator. it does not have components as a del operator(so it must not be multiplied as i and i & j and j then k and k) ........sir please tell me it is correct or not........
ReplyDeletethank u sir for making us able to interact with u via such a blog
ReplyDeletesir abhi thik thik samajh me aa raha hai
This comment has been removed by the author.
ReplyDeleteYes the laplacian is a scalar operator which is the sum of the three second partial derivatives. It can operate on both scalar and vector fields. But remember that its also a differential operator like d/dx so it satisfies all the properties of a differential operator like d/dx.
ReplyDeleteThe Laplacian can come from div ( grad of a scalar field) or it can also directly operate on each component of a vector field. ( remember that ecah component of a vector field is a scalar field in general.)
Sumit Gupta posted a comment on vectors in Facebook
ReplyDeleteas to how they are transformation dependent.
My response: It depends on the transformation. A quantity is a vector under a
given set of transformations. The same quantity may be a vector under one set of transformations but may be a rank two tensor under another set of transformation.
An example is ...the Electric Field. You will learn within a year that
E is a vector under rotations but is not a vector under Lorentz Transformations in Special Theory of Relativity.
Sumit Gupta said: well can you PLEASE give me an example such that which doesnt behave as a vector under transformations but satisfies our earlier definition of magnitude and direction?
ReplyDeleteMy response: Such an example can be cooked up but is not mathematically interesting. Griffiths has an example of this sort Sec 1.1.5 #rd Edition where he exlains how the concept of a direction is rather vague and dependent on the coordinate system.
The t...ransformation law actually specifies what is meant by direction in a coordinate independent fashion. This is the key.
So magnitude and direction are fine but one has to clearly specify what is meant by "direction" as the term is coordinate system dependent.
Since you are pondering let me go a little ahead and state that
there are two ways of understanding a vector. One is algebraic
in the sense of something called a "Vector Space"
( you will learn this in Math 102 Linear Algebra) and tge other is geometrically by knowing that they are actually "tensors" of rank 1 under a given set of coordinate transformations. ( A Scalar is tensor of rank 0 as i explained in my lectures.)
sir due to the preparations for freshers i m not getin time to read or complete d assignments.....plz sir tell me how to do with it.....will takin part in freshers affect my study????plz sir help!!!!!!
ReplyDeletei m follow' the last 4 classes well but initial lecture on transformation and it s application. r a bit difficult. to handle
ReplyDeleteTaking part in Freshers Intro will not affect your studies. Just put in some time for the problems. Managing time and a sense of balance between study and extra curricular activities is a must in IIT.
ReplyDeleteDont worry about transformations. If you wish you can come and talk to me about your difficulties after the class or with an appointment.
ReplyDeleteis there a precise way to describe vector fields in space with their magnitude and direction, field lines represent the direction only and arrows cannot be used to give the magnitude of vectors at every point, i guess. i read somewhere about contour maps but couldn't get it, are they of any help?
ReplyDeleteBoth arrows and field lines are reasonably precise.
ReplyDeleteBoth the pictures can be simulated by a computer
program to provide the direction and relative field strengths everywhere. In the field line picture the density of lines provide a measure for the relative strength. This is for visualisation.
The most precise way is mathematical through the transformation law. The exact transformation law for vector field is beyond the scope of this course.
Contour maps are usually for scalar fields. For example a system of isotherms is a contour map
for the temperature function.
I dont know if contour maps can be also used to
visualize vector fields. You have to catch a computer guy. I am a physicist :-)
sir, i always confuse with the difference between vector and vector field. and why curl of electric field is zero.
ReplyDeleteA Vector Field is a Vector function of co-ordinates
ReplyDeletelike A (x, y,z) having 3 components each of which is
a scalar function. So a Vector field has different magnitude and different direction at every point in
space. An ordinary vector is usually refered to as a
constant vector that is both its direction and magnitude are fixed. Vector fields transform in a
very different way than ordinary constant vectors under rotations.
You have proved today in DIPA that Curl ( grad φ=0 when φ is a scalar field. The electric field maybe expressed as the -grad V where V is the electric potential. So Curl E is Curl grad V and that is zero as V is a scalar field. We will do this in class shortly.
Sir
ReplyDeleteI am facing problems in understanding first boundary theorem for the gradient of the scalar field. Also please do give an example of the line integral used in physics which is path dependent.
With regards
Firstly please comment on the correct recent post
ReplyDeleteso that its easy for other people to join in. I have made a post on the last lecture so it would be best to comment on that.
The first boundary theorem is the easiest and basically a generalization of one variable integral of df/dx dx from a to b is f(b)-f(a).
so integral of grad phi ( x, y,z)dot dl= integral d phi= phi(b)-phi:(a) thats all. Nothing more to understand.
If you are still having problems then see me after the class.
Path dependent line integrals occur in advanced physics issues and are not necessary at your level.
is it advantegeous for us to attempt questions from griffiths?
ReplyDeleteKind of late in asking this question. Grifiths examples related to the matter in the lectures and some of the problems in the text matter itself and a few simple problems from the Chapter end sets. Most of the problems in IPSA and DIPA are from Griffiths and they are chosen with special care to highlight the issues in the lecture. DIPA/IPSA problems are the key to the course and reading Griffiths with the help of this blog after the lectures.
ReplyDelete