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arivero
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I'd like to head about how people visualizes a cotangent vector, ie a differential form. I like to use surfaces of f(x)=cte.
shankarvn said:Does this have anything to do with the contravariant and covariant representations of a vector? Can u explain the antennae example a bit more. That seems to be a nice way to understand. How are these differential forms related to the "concept of an infinitesimal ie. the way we understand things like dx,dy,dz etc..." ...??Since we integrate on differential forms, I believe there has to be a relation to the way we understand dx,dy etc as something very close to zero but not zero. Can someone explain?
Shankar
shankarvn said:Hi quetzalcoatl9
Thanks a lot..That really helped ... I just have one last question. In this antennae example are we not assuming that the field varies linearly??Isn't that a sweeping assumption ??I don't know much about EM theory and maxwell's eqns..But I was just wondering whether we know apriori that the field varies linearly.
Thanks again
Shankar
shankarvn said:Hi quetzalcoatl9
This might not make sense but I still wanted to ask. Can we say that at a pont in the manifold [tex]TM_p [/tex] , the tangent vector has a covariant representation [tex]dx_1[/tex] and a contravariant component [tex]dx^1[/tex].
I was taught that we can think of 2 representations(components) of a vector with respect to a basis and with respect to its dual(reciprocal basis). So are we looking at the tangent vector with respect to its components, with respect to its local basis(call it covariant) and with respect to its reciprocal/dual basis(call it its contravariant comp) . This question is due to the fact that I do not understand the concept of a differential map between tangent spaces. This is because they write out the differential map(Jacobian) between tangent spaces [tex]R^M_{p}\rightarrow R^N_{f(p)} [/tex] and that happens to be the map between the differential forms(cotangent vectors) from what we know " as the way differentials map with respect to coordinate transformation". This kind of forces me to think that cotangent vectors and tangent vectors are kind of representations of the same thing..If you feel I am talking absolute nonsense please ignore this question(do tell me you are ignoring it ).
Thanks
Shankar
A contangent vector is a mathematical concept commonly used in differential geometry. It is a vector that represents the rate of change of a function with respect to its independent variables. In simpler terms, it is a vector that describes the direction and magnitude of change of a function at a specific point.
Visualization of contangent vectors helps us understand the behavior of a function at a specific point. It provides a visual representation of the direction and magnitude of change, which can aid in understanding the overall behavior of the function and its relationship to its independent variables.
Contangent vectors can be visualized as arrows on a graph, with the direction and length of the arrow representing the direction and magnitude of change at a specific point. They can also be represented as a set of coordinates in a coordinate system.
Visualizing contangent vectors can be useful in various fields such as physics, engineering, and computer graphics. It can help in understanding the behavior of physical systems, designing efficient structures, and creating realistic computer simulations.
To improve understanding, one can practice visualizing contangent vectors in different scenarios and with various functions. It can also be helpful to study the underlying mathematical principles and concepts, such as differential calculus, that govern the behavior of contangent vectors.