@empty Teach me vector calculus pleaseeeee

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Stokes!
Stokes!
Haha sure, I just don't know where to begin, what do you already know?

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

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