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Astrophysics

  • one year ago

@empty Teach me vector calculus pleaseeeee

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  1. Astrophysics
    • one year ago
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    Stokes!

  2. Astrophysics
    • one year ago
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    Stokes!

  3. Empty
    • one year ago
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    Haha sure, I just don't know where to begin, what do you already know?

  4. Astrophysics
    • one year ago
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    I know how to calculate stuff, just have a hard time understanding the theory, like I know Stokes is Green's in higher dimensions, but when I deal with triple integrals, I just don't understand it...

  5. Empty
    • one year ago
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    Unfortunately I think my understanding of these theorems sorta really comes through in learning tensor calculus, because then you can truly see how green's theorem, gauss' theorem, and stokes' theorem are all identical.

  6. Astrophysics
    • one year ago
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    Towards the end of vec calc is where it gets confusing haha

  7. Astrophysics
    • one year ago
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    Actually I can figure it out, long as I have a good understanding of curl and divergence, so...

  8. Empty
    • one year ago
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    Ahh ok well those are something I can help you with too.

  9. Astrophysics
    • one year ago
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    They usually give a fluid analogy, but what I kind of think of when curl is mentioned is, angular momentum haha..

  10. Astrophysics
    • one year ago
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    No actually it would be more related to torque!

  11. Empty
    • one year ago
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    Well possibly linear algebra could help, but I don't think that it will help in visualizing a vector field. I guess when I think of taking the curl of a vector field I imagine that the new vector field I'm given shows the axis of maximum torque if you were there. It's hard to describe but in that sense you can sorta think of the curl as the gradient of a scalar field.

  12. Astrophysics
    • one year ago
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    Ok yeah, I can follow all of this actually as I understand the mathematical definitions, but the vector field is the culprit haha

  13. Empty
    • one year ago
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    I think the best vector fields to understand are conservative vector fields since these come up very naturally in physics. I think before we talk about vector fields, let's talk about scalar fields to build a strong foundation. Can you give an example of some scalar or vector fields you know of or ones you're unsure of?

  14. Astrophysics
    • one year ago
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    Well this isn't a example, but I know Newton's law of gravitation would be a vector field/ also Coloumb's law, and a scalar field is gradient fields right?

  15. Astrophysics
    • one year ago
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    So a scalar field would be a function of two variables

  16. Astrophysics
    • one year ago
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    And these laws are all conservative fields

  17. Astrophysics
    • one year ago
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    Or in simple terms, scalar field I think about magnitudes while vector field gives us the direction, but in 3d it's kind of confusing haha, I'm not very good at drawing them either, it's sort of hard

  18. Empty
    • one year ago
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    No not necessarily, a scalar field just means if we take some point in space, we can evaluate it and get a scalar value. So a good example might be temperature can be seen as a scalar field, that's usually the most obvious one. Another scalar field is potential energy, since energy is not a vector. We can also think of your computer screen as a scalar field with every point in the xy plane evaluating to its frequency, which is a specific color. I am sure there are other scalar fields we could come up with too.

  19. Empty
    • one year ago
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    Another scalar field we could imagine would be air pressure.

  20. Astrophysics
    • one year ago
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    Haha, so the magnitude

  21. Empty
    • one year ago
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    But in order to have a conservative vector field (shortened to just conservative field since we don't really have conservative scalar fields) we must start out with a scalar field to begin with. That is to say, every conservative field has a corresponding scalar field.

  22. Empty
    • one year ago
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    Hmmm well maybe, depends, magnitude of what?

  23. Astrophysics
    • one year ago
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    That's a good way you put it, so what I mean by magnitude is you can figure out the temperature for example anywhere in space but we won't know the "direction" of the temperature.

  24. Astrophysics
    • one year ago
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    A while back I did read the definition of divergence via pauls online notes, and I think scalar field is related to the divergence as he used an analogy with a microwave haha

  25. Astrophysics
    • one year ago
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    So that is why I said gradient field

  26. Empty
    • one year ago
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    Well it's not so much that the temperature in this model has direction. Although technically temperature is the movement of particles, we're sorta throwing that idea away. In this sense the gradient of the temperature field will be the direction at any point at which to go to the greatest increase. You can imagine that you don't need calculus to define this concept. You can imagine that in space there is a unique vector at every point in a temperature field that feels the area next to it and then points to the place where it's warmest nearby it.

  27. anonymous
    • one year ago
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    I think Stokes is \[ \oint_{\partial S}\mathbf f \cdot d\mathbf r = \iint_{S} \nabla \times \mathbf f\cdot d\mathbf S \]Where \(\mathbf r\) is a parametrization of the closed boundary of the surface \(S\), and \(\mathbf S\) is a parametrization of \(S\).

  28. Empty
    • one year ago
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    Yeah, the divergence of a scalar field is always a conservative vector field, that's the definition.

  29. Astrophysics
    • one year ago
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    Right, I think I'm understanding this now haha, and yes wio that is the definition, it's just understanding the concept that's what's troubling!

  30. anonymous
    • one year ago
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    The idea is that the curl will make the vector field easier to deal with.

  31. Empty
    • one year ago
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    I think it might be best to see if you understand Green's theorem first, since it's the same thing. What do you understand about Green's theorem @Astrophysics ?

  32. Astrophysics
    • one year ago
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    I could give you the calculus type definition of it, as in it's a relationship between a line integral around a closed curve C...but I could not exactly tell you what applications it would be useful for.

  33. Astrophysics
    • one year ago
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    potential function

  34. Astrophysics
    • one year ago
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    As you see vector calc was not my strongest haha.

  35. Empty
    • one year ago
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    Ahhh ok well then let's not think of "Greens" theorem, let's directly consider it as a special case of Stokes' theorem so that once we understand it, then we can say the higher dimensional version is really just the same idea, we were just looking at one piece at a time. So in order to do that, I guess I'll have to draw some pictures. But I think we should really understand what a line integral is first before we do this, so do you understand what a line integral is or an application for it, just sorta throw out whatever information however random it is, that you know about line integrals. :P

  36. anonymous
    • one year ago
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    If you have a closed circuit, then you could possibly use stokes theorem to calculate such a line integral.

  37. Astrophysics
    • one year ago
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    Mhm ok, there's a lot to line integrals, and definitions. So the most basic thing I can say about them is, we use them to integrate over a curve rather than certain intervals, which also require parameters and such.

  38. anonymous
    • one year ago
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    It's in Maxwell's equations

  39. Astrophysics
    • one year ago
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    Yeah, a lot of vector calculus is in E&M

  40. Astrophysics
    • one year ago
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    If not all

  41. anonymous
    • one year ago
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    Do you understand what a vector field is?

  42. Empty
    • one year ago
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    @wio Have you been reading this thread?

  43. anonymous
    • one year ago
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    Yes

  44. anonymous
    • one year ago
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    But I'm still not sure if he knows what a vector field is.

  45. Astrophysics
    • one year ago
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    I think so man, it's like in R^3 and can be expressed in component functions and stuff

  46. anonymous
    • one year ago
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    Actually it doesn't need to be in \(\mathbb R^3\).

  47. anonymous
    • one year ago
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    You can have a vector field in \(\mathbb R\), but it's be \(1\) component vectors.

  48. Astrophysics
    • one year ago
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    Right!

  49. anonymous
    • one year ago
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    You probably already know this, but a scalar function will associate with point in the coordinate system (in its domain) some scalar value, while a vector field will associate the point with a vector.

  50. anonymous
    • one year ago
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    When we graph scalar functions, we usually either use level curves, or we add in a new coordinate whose value will represent the function's value. When we graph a vector function, we draw a vector pointing in the proper direction, and we try to make it's length somewhat representative of its magnitude. There is no way we could use an added coordinate to represent both magnitude and direction of a vector. When you think about it, we treat vector fields very different graphically, but at the end of the day, it's just a function that outputs vectors.

  51. anonymous
    • one year ago
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    Do path integrals make sense to you? That is integrating a scalar function over a curve?

  52. Astrophysics
    • one year ago
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    I think I understand a decent amount, but I'm still reading about them. I've done problems with them, but I'm still trying to process the meaning of it, so I'm reading about it right now actually

  53. anonymous
    • one year ago
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    Okay, if I gave you \[ \int_0^1 x^2 ~ dx \]How would you understand it?

  54. anonymous
    • one year ago
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    Maybe we can leverage that, since I'm guessing it makes sense to you what it means.

  55. Astrophysics
    • one year ago
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    As in |dw:1436000594348:dw|

  56. anonymous
    • one year ago
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    So you interpret it as area under the curve.

  57. Astrophysics
    • one year ago
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    Yup

  58. anonymous
    • one year ago
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    What is \(y\)?

  59. anonymous
    • one year ago
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    The integral I gave you only had \(x\), and your graph has a \(y\), so I want you to tell me what's up with that.

  60. anonymous
    • one year ago
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    Are you not here or do you not know why or what?

  61. Astrophysics
    • one year ago
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    Hey, ya sorry wio I'm actually just doing some green theorem and understanding line integrals so I wasn't really paying attention

  62. anonymous
    • one year ago
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    oh ok

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