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anonymous
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ques
Mathematics
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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anonymous
  • anonymous
what is the question
anonymous
  • anonymous
From Stoke's Theorem we have \[\iint_\limits S (\vec \nabla \times \vec F). \hat n \space ds=\oint_\limits C \vec F.d \vec r\] Where S is any surface and C is a closed curve around the surface I can see how this works out in the case of a 2D surface |dw:1443885818098:dw| But if we consider the surface of a 3D object, we have |dw:1443885892309:dw| Here S is the lateral surface of that figure What do we take for our C here??
ganeshie8
  • ganeshie8
C is the closed curve which is the boundary of that surface |dw:1443886311804:dw|

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anonymous
  • anonymous
It isn't really enclosing the surface though??
anonymous
  • anonymous
Oh, it must be bounded by C that's the only condition
anonymous
  • anonymous
Suppose we want to find the surface integral of curl F both through the lateral surface and the circular surface, Would that be done something like.. \[\oint_\limits C \vec F.d \vec r+\iint_\limits R (\vec \nabla \times \vec F).\hat k \space dxdy\]|dw:1443886644120:dw|
IrishBoy123
  • IrishBoy123
the line integral and the surface integral are the same, that is what Stokes' theorem says
IrishBoy123
  • IrishBoy123
you know the [deformable] butterfly net analogy?
anonymous
  • anonymous
Nope, but is my step correct for finding the surface integral of curl F through the whole surface(including circular top) |dw:1443887053149:dw|
IrishBoy123
  • IrishBoy123
yes but |dw:1443887366725:dw|
anonymous
  • anonymous
Hm, that's easy to understand, because the curves bounding these surfaces are the same, so the line integral through these curves should be the same for a particular vector field!
IrishBoy123
  • IrishBoy123
and the flat net enclosed by the ring = the puffed out nets because all are enclosed by same ring.....and covered by the same line integral
anonymous
  • anonymous
Wait does that mean for total surface we have, \[2 \times \oint_\limits{C} \vec F . d \vec r\] ??
IrishBoy123
  • IrishBoy123
at the risk of sounding agricultural, it comes in the hoop and goes out the net.
IrishBoy123
  • IrishBoy123
i can post an example that can be done several ways
anonymous
  • anonymous
So it will be 2 times the line integral??? for total surface??
anonymous
  • anonymous
If it's equal for the surfaces, since they have the same C
anonymous
  • anonymous
then it just becomes 2 times for total surface?
IrishBoy123
  • IrishBoy123
no, they should net out to zero so take a sphere |dw:1443888358128:dw|
IrishBoy123
  • IrishBoy123
you should be able to do both surfaces using the loop so they are the same but they cancel out |dw:1443888386190:dw||dw:1443888398064:dw|
anonymous
  • anonymous
|dw:1443888517480:dw|
anonymous
  • anonymous
Applying stokes theorem on R and S, they convert to line integrals of the same curve, so just 2 times the line integral for total??
anonymous
  • anonymous
Why do they cancel out instead of adding up??
IrishBoy123
  • IrishBoy123
because in the physical sense you are typically looking at the flux (like flow) of the curl of some field. so say from Maxwell you have \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}\) relating the curl of an eletric field with a magnetic field and its flux.
anonymous
  • anonymous
Hmmm, I remember something similar from school \[e.m.f=-\frac{d \Phi_{B}}{dt}\] Faraday's law of electromagnetic induction
anonymous
  • anonymous
Your equation looks different
IrishBoy123
  • IrishBoy123
yours is faraday's in simplest form - induced voltage from a changing magnetic field i quoted maxwell #3 in its differential form which is the basis of what you quoted my point though is that when we look at surfaces we are usually concerned with the rate at which something is flowing through it, be it a fluid or a magnetic field.
IrishBoy123
  • IrishBoy123
scroll down here to equation 7 http://www.maxwells-equations.com/faraday/faradays-law.php
anonymous
  • anonymous
Shouldn't it be a double integral on the right side of equation 7
IrishBoy123
  • IrishBoy123
there's a lot of variation in notation. i think the \(\int_S \dots dS \) is enough for some people to signify a double integral
anonymous
  • anonymous
Oh, cuz I thought about it like this \[\vec \nabla \times \vec E=-\frac{\partial \vec B}{\partial t}\] \[(\vec \nabla \times \vec E).d \vec A=-\frac{\partial \vec B}{\partial t}.d \vec A\]\[\iint (\vec \nabla \times \vec E). \space d \vec A=-\iint \frac{\partial \vec B}{\partial t}. \space d \vec A\]\[\oint \vec E.d \vec l=-\iint \frac{\partial \vec B}{\partial t}.d \vec A\]
anonymous
  • anonymous
Electromagnetism is fascinating, I can't wait to study it in later semesters, the moment when you realize ahh you can apply the math you've learned so beautifully
IrishBoy123
  • IrishBoy123
that's pretty good!
anonymous
  • anonymous
I think I'll leave discussion about finding total flux over that surface, it still looks a bit confusing and I doubt I'd use it right now, most of the problems require me to find the flux over the lateral surfaces (they mainly want to see how u can apply the theorems etc.) so my exams will be more theorems(statements) and problems based. My main doubt was how to find C for a 3D figure I quote from my syllabus "The emphasis of course is on applications in solving problems of interest to physicists. The students are to be examined entirely on the basis of problems, seen and unseen."
IrishBoy123
  • IrishBoy123
try this. i could really do with the practise too but i've got a lot of rugby to watch this evening:-) so i'll have a go later.... it is set up to be do-able in a few ways, so might be interesting.
1 Attachment
IrishBoy123
  • IrishBoy123
oh, i mean example 1
anonymous
  • anonymous
\[\vec V=4y \hat i+x \hat j+2z \hat k\] S is \[x^2+y^2+z^2=a^2\] \[z \ge 0\] |dw:1443891158189:dw| Hmm one way we can do is convert into line integral using stokes' theorem! \[\iint_\limits{S}(\vec \nabla \times \vec V).\hat n \space d \sigma=\oint_\limits C \vec V.d \vec r\] C is the circular loop in the xy plane therefore z=0, dz=0 \[\iint_\limits S=\oint_\limits{C}4ydx+xdy=\int\limits_{0}^{2\pi} (-4a^2\sin^2(\theta)+a^2\cos^2(\theta)) d \theta\]\[\iint_\limits{S}=\frac{a^2}{2}\int\limits_{0}^{2\pi}(5\cos(2\theta)-3)d \theta=\frac{a^2}{2}[\frac{5\sin(2 \theta)}{2}-3 \theta]_{0}^{2\pi}=-3 \pi a^2\] 2nd we can find curl then take the projection on xy plane \[\iint_{S}=\iint_{R}(\vec \nabla \times \vec V).\hat n \frac{dxdy}{\hat n . \hat k}\] \[\vec \nabla \times \vec V=-3 \hat k\] \[\hat n=\frac{x \hat i+y \hat j+z \hat k}{\sqrt{x^2+y^2+z^2}}\]\[(\vec \nabla \times \vec V).\hat n=-\frac{3z}{\sqrt{x^2+y^2+z^2}}\]\[\frac{1}{\hat n . \hat k}=\frac{\sqrt{x^2+y^2+z^2}}{z}\] \[\iint_\limits{S}=\iint_\limits{R}-\frac{3z}{\sqrt{x^2+y^2+z^2}}.\frac{\sqrt{x^2+y^2+z^2}}{z}dxdy=-3\iint_\limits{R}dxdy=-3\pi a^2\]
anonymous
  • anonymous
IrishBoy123
  • IrishBoy123
yes! i think this is a really good book, you will know the maths already: http://www.amazon.co.uk/A-Students-Guide-Maxwells-Equations/dp/0521701473 there is a pdf somewhere on the net so you can try it first and see if you like it. by the time you finish it you will see how Maxwell worked out the speed of light whilst sitting at his desk doing some maths! amazing stuff really.

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