Question

@empty Teach me vector calculus pleaseeeee

Answer 1

Stokes!

Answer 2

Stokes!

Answer 3

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

Answer 4

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...

Answer 5

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.

Answer 6

Towards the end of vec calc is where it gets confusing haha

Answer 7

Actually I can figure it out, long as I have a good understanding of curl and divergence, so...

Answer 8

Ahh ok well those are something I can help you with too.

Answer 9

They usually give a fluid analogy, but what I kind of think of when curl is mentioned is, angular momentum haha..

Answer 10

No actually it would be more related to torque!

Answer 11

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.

Answer 12

Ok yeah, I can follow all of this actually as I understand the mathematical definitions, but the vector field is the culprit haha

Answer 13

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?

Answer 14

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?

Answer 15

So a scalar field would be a function of two variables

Answer 16

And these laws are all conservative fields

Answer 17

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

Answer 18

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.

Answer 19

Another scalar field we could imagine would be air pressure.

Answer 20

Haha, so the magnitude

Answer 21

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.

Answer 22

Hmmm well maybe, depends, magnitude of what?

Answer 23

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.

Answer 24

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

Answer 25

So that is why I said gradient field

Answer 26

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.

Answer 27

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 \(S\).

Answer 28

Yeah, the divergence of a scalar field is always a conservative vector field, that's the definition.

Answer 29

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!

Answer 30

The idea is that the curl will make the vector field easier to deal with.

Answer 31

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 ?

Answer 32

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.

Answer 33

potential function

Answer 34

As you see vector calc was not my strongest haha.

Answer 35

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

Answer 36

If you have a closed circuit, then you could possibly use stokes theorem to calculate such a line integral.

Answer 37

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.

Answer 38

It's in Maxwell's equations

Answer 39

Yeah, a lot of vector calculus is in E&M

Answer 40

If not all

Answer 41

Do you understand what a vector field is?

Answer 42

@wio Have you been reading this thread?

Answer 43

Yes

Answer 44

But I'm still not sure if he knows what a vector field is.

Answer 45

I think so man, it's like in R^3 and can be expressed in component functions and stuff

Answer 46

Actually it doesn't need to be in \(\mathbb R^3\).

Answer 47

You can have a vector field in \(\mathbb R\), but it's be \(1\) component vectors.

Answer 48

Right!

Answer 49

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.

Answer 50

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.

Answer 51

Do path integrals make sense to you? That is integrating a scalar function over a curve?

Answer 52

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

Answer 53

Okay, if I gave you \[
\int_0^1 x^2 ~ dx
\]How would you understand it?

Answer 54

Maybe we can leverage that, since I'm guessing it makes sense to you what it means.

Answer 55

As in