Astrophysics
  • Astrophysics
@empty Teach me vector calculus pleaseeeee
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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Astrophysics
  • Astrophysics
Stokes!
Astrophysics
  • Astrophysics
Stokes!
Empty
  • Empty
Haha sure, I just don't know where to begin, what do you already know?

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Astrophysics
  • Astrophysics
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...
Empty
  • Empty
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.
Astrophysics
  • Astrophysics
Towards the end of vec calc is where it gets confusing haha
Astrophysics
  • Astrophysics
Actually I can figure it out, long as I have a good understanding of curl and divergence, so...
Empty
  • Empty
Ahh ok well those are something I can help you with too.
Astrophysics
  • Astrophysics
They usually give a fluid analogy, but what I kind of think of when curl is mentioned is, angular momentum haha..
Astrophysics
  • Astrophysics
No actually it would be more related to torque!
Empty
  • Empty
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.
Astrophysics
  • Astrophysics
Ok yeah, I can follow all of this actually as I understand the mathematical definitions, but the vector field is the culprit haha
Empty
  • Empty
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?
Astrophysics
  • Astrophysics
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?
Astrophysics
  • Astrophysics
So a scalar field would be a function of two variables
Astrophysics
  • Astrophysics
And these laws are all conservative fields
Astrophysics
  • Astrophysics
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
Empty
  • Empty
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.
Empty
  • Empty
Another scalar field we could imagine would be air pressure.
Astrophysics
  • Astrophysics
Haha, so the magnitude
Empty
  • Empty
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.
Empty
  • Empty
Hmmm well maybe, depends, magnitude of what?
Astrophysics
  • Astrophysics
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.
Astrophysics
  • Astrophysics
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
Astrophysics
  • Astrophysics
So that is why I said gradient field
Empty
  • Empty
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.
anonymous
  • anonymous
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 \(S\).
Empty
  • Empty
Yeah, the divergence of a scalar field is always a conservative vector field, that's the definition.
Astrophysics
  • Astrophysics
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!
anonymous
  • anonymous
The idea is that the curl will make the vector field easier to deal with.
Empty
  • Empty
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 ?
Astrophysics
  • Astrophysics
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.
Astrophysics
  • Astrophysics
potential function
Astrophysics
  • Astrophysics
As you see vector calc was not my strongest haha.
Empty
  • Empty
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
anonymous
  • anonymous
If you have a closed circuit, then you could possibly use stokes theorem to calculate such a line integral.
Astrophysics
  • Astrophysics
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.
anonymous
  • anonymous
It's in Maxwell's equations
Astrophysics
  • Astrophysics
Yeah, a lot of vector calculus is in E&M
Astrophysics
  • Astrophysics
If not all
anonymous
  • anonymous
Do you understand what a vector field is?
Empty
  • Empty
@wio Have you been reading this thread?
anonymous
  • anonymous
Yes
anonymous
  • anonymous
But I'm still not sure if he knows what a vector field is.
Astrophysics
  • Astrophysics
I think so man, it's like in R^3 and can be expressed in component functions and stuff
anonymous
  • anonymous
Actually it doesn't need to be in \(\mathbb R^3\).
anonymous
  • anonymous
You can have a vector field in \(\mathbb R\), but it's be \(1\) component vectors.
Astrophysics
  • Astrophysics
Right!
anonymous
  • anonymous
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.
anonymous
  • anonymous
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.
anonymous
  • anonymous
Do path integrals make sense to you? That is integrating a scalar function over a curve?
Astrophysics
  • Astrophysics
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
anonymous
  • anonymous
Okay, if I gave you \[ \int_0^1 x^2 ~ dx \]How would you understand it?
anonymous
  • anonymous
Maybe we can leverage that, since I'm guessing it makes sense to you what it means.
Astrophysics
  • Astrophysics
As in |dw:1436000594348:dw|
anonymous
  • anonymous
So you interpret it as area under the curve.
Astrophysics
  • Astrophysics
Yup
anonymous
  • anonymous
What is \(y\)?
anonymous
  • anonymous
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.
anonymous
  • anonymous
Are you not here or do you not know why or what?
Astrophysics
  • Astrophysics
Hey, ya sorry wio I'm actually just doing some green theorem and understanding line integrals so I wasn't really paying attention
anonymous
  • anonymous
oh ok

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