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Astrophysics
 one year ago
@empty Teach me vector calculus pleaseeeee
Astrophysics
 one year ago
@empty Teach me vector calculus pleaseeeee

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Empty
 one year ago
Best ResponseYou've already chosen the best response.2Haha sure, I just don't know where to begin, what do you already know?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1I 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
 one year ago
Best ResponseYou've already chosen the best response.2Unfortunately 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
 one year ago
Best ResponseYou've already chosen the best response.1Towards the end of vec calc is where it gets confusing haha

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Actually I can figure it out, long as I have a good understanding of curl and divergence, so...

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Ahh ok well those are something I can help you with too.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1They usually give a fluid analogy, but what I kind of think of when curl is mentioned is, angular momentum haha..

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1No actually it would be more related to torque!

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Well 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
 one year ago
Best ResponseYou've already chosen the best response.1Ok yeah, I can follow all of this actually as I understand the mathematical definitions, but the vector field is the culprit haha

Empty
 one year ago
Best ResponseYou've already chosen the best response.2I 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
 one year ago
Best ResponseYou've already chosen the best response.1Well 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
 one year ago
Best ResponseYou've already chosen the best response.1So a scalar field would be a function of two variables

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1And these laws are all conservative fields

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Or 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
 one year ago
Best ResponseYou've already chosen the best response.2No 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
 one year ago
Best ResponseYou've already chosen the best response.2Another scalar field we could imagine would be air pressure.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Haha, so the magnitude

Empty
 one year ago
Best ResponseYou've already chosen the best response.2But 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
 one year ago
Best ResponseYou've already chosen the best response.2Hmmm well maybe, depends, magnitude of what?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1That'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
 one year ago
Best ResponseYou've already chosen the best response.1A 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
 one year ago
Best ResponseYou've already chosen the best response.1So that is why I said gradient field

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Well 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
 one year ago
Best ResponseYou've already chosen the best response.0I 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
 one year ago
Best ResponseYou've already chosen the best response.2Yeah, the divergence of a scalar field is always a conservative vector field, that's the definition.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Right, 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
 one year ago
Best ResponseYou've already chosen the best response.0The idea is that the curl will make the vector field easier to deal with.

Empty
 one year ago
Best ResponseYou've already chosen the best response.2I 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
 one year ago
Best ResponseYou've already chosen the best response.1I 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
 one year ago
Best ResponseYou've already chosen the best response.1potential function

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1As you see vector calc was not my strongest haha.

Empty
 one year ago
Best ResponseYou've already chosen the best response.2Ahhh 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
 one year ago
Best ResponseYou've already chosen the best response.0If you have a closed circuit, then you could possibly use stokes theorem to calculate such a line integral.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Mhm 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
 one year ago
Best ResponseYou've already chosen the best response.0It's in Maxwell's equations

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Yeah, a lot of vector calculus is in E&M

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Do you understand what a vector field is?

Empty
 one year ago
Best ResponseYou've already chosen the best response.2@wio Have you been reading this thread?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0But I'm still not sure if he knows what a vector field is.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1I think so man, it's like in R^3 and can be expressed in component functions and stuff

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Actually it doesn't need to be in \(\mathbb R^3\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You can have a vector field in \(\mathbb R\), but it's be \(1\) component vectors.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You 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
 one year ago
Best ResponseYou've already chosen the best response.0When 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
 one year ago
Best ResponseYou've already chosen the best response.0Do path integrals make sense to you? That is integrating a scalar function over a curve?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1I 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
 one year ago
Best ResponseYou've already chosen the best response.0Okay, if I gave you \[ \int_0^1 x^2 ~ dx \]How would you understand it?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Maybe we can leverage that, since I'm guessing it makes sense to you what it means.

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1As in dw:1436000594348:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So you interpret it as area under the curve.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0The 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
 one year ago
Best ResponseYou've already chosen the best response.0Are you not here or do you not know why or what?

Astrophysics
 one year ago
Best ResponseYou've already chosen the best response.1Hey, ya sorry wio I'm actually just doing some green theorem and understanding line integrals so I wasn't really paying attention
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