maverick_starstrider said:
We're the sorry the universe does not obey laws that are sufficiently accessible to you. Take it up with your local deity of choice.
If you've got an issue with the poster, use the report button.
rockyshephear said:
But I thought the definition of the dot product IS |A||B| cos theta times some unit vector.
Math often presents us with several related definitions. This definition is given to most students first, because we all knows what an angle is, and it's easy to see that the dot product is related to the angle two vectors meet.
However, the inner product can also be defined in terms of orthogonal projection. It's something like one vector casting a "shadow" over another (to use fluffy, visual terms). Inner products can also be defined on other kinds of "vectors". Later on in physics, you often deal with function spaces -- sets where each "vector" is actually a complex-valued function. Unlike vectors in R^3, functions don't have a direction and don't meet at angles. But we can still define an inner product on them which preserves many useful properties. (The inner product between two functions is usually defined as the integral of the product of the two functions).
So electric field is what it is without any charges dropped in and the flux is the interaction after a charge is dropped in? So without test charge there is a field but no flux. Once you throw in a charge, the you have flux?
The electric field is what it is. The test charges are only there to show you the "shape" of the field. After you understand how the field looks, forget everything you know about test charges. Just understand that the field can point one way, and the particle might even move backwards! (If the charge on the particle is negative)
One way to think of test charges is like this. Imagine the field is invisible (because... it actually is...) Now, your test charge is like a microscope. Wherever you hold the test charge, you are allowed to see the field at that point in space. Take the test charge away, and the field is invisible, but it's still there.
There is no "flux" until you pick a surface. You have to have a field AND a surface, and then you talk about flux
of the field
through the surface.
Ok. Can I claim semantics on this one?
Take a silly example. I have the vector <3, 4, 0>. It's length is 5. What is it's angle? There's no good answer. You have to take the angle with respect to another vector... and with respect to a given plane (... and orientation on top of that!) Granted the notion of "direction" isn't great either. But intuitively, "angle" is more misleading.
Your brain becoming fried is probably an indication you have learned something. I remember struggling with these ideas when I started too. Give your mind some rest, and when you're showering or ready to go to bed, it will just sort of click. Replay the definitions of everything and the visual examples in your head, and you'll get it eventually.
I wonder do all you guys have visualizations of each concept or do you think only in term of the equations?
Visualizations are a huge part of a lot of math. The mind works a great deal better in two dimensional images than it does in symbols on a page. The hard work you have to do is be able to translate back and forth between them. It's almost like reading and writing sheet music or tabs. A tab doesn't sound at all like the music. And the music doesn't look like a tab. But the two are in some way the same. Part of the job of a musician is learning to translate between the two.
Whenever you see an integral or a direction derivative or a vector, think back to the simplest examples. There's a joke that goes like this:
A physicist and a mathematician go to a string theory conference. After the conference, the physicist exclaims "This nonsense about 9-dimensional spacetime is impossible to visualize!" The mathematician turns to him and says "Oh, it's not so hard." The physicist asks him, "Well, how do
you do it then?" The mathematicians explains, "It's simple, I just imagine n-dimensional space, then set n equal to 9".
Of course, no one can really "see" 9-dimensional space, much less "n-"dimensional space. But the basic ideas in n-dimensional space are the same as they are in 3. There are vectors. They have lengths. You can take the dot product between any two. You can rotate any vector in any plane by any angle. You can add vectors in the same way. You can scale them in the same way. They are very similar.