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1. Notation question - R^m -> r^n

We can write \begin{align} f:\mathbb{R}^m&\to\mathbb{R}^n\\ x&\mapsto y=f(x) \end{align} and \begin{align} y&=(y_1,\dots,y_n)\\ &=f(x)\\ &=f(x_1,\dots,x_m)\\ &=(f_1(x),\dots,f_n(x))\\ &=(f_1(x_1,\dots,x_m),\dots,f_n(x_1,\dots,x_m)). \end{align} Try drawing the graphs of the functions...
2. Any real world use of imaginary numbers?

Let's follow the idea that the numbers don't exist. Perhaps that is true. You can view it as being the algebraic completion of the real numbers. You can create methods which work very generally. Often, these methods give you a very general form of the solution, and a couple steps later, you...
3. Free R-modules

Can we think of it as $$F\cong\prod_{\alpha\in J}R_\alpha$$ This is the underlying abelian group (analogous to vectors in vector space), and it looks like there is a natural way to multiply on the left by elements of R (analogous to scalars in a vector space). For F above, a basis could be...
4. PDE and heat equation

I would associate it with the shape of the distribution. It is the second derivative, so what does this mean for a sine function? A quadratic equation? A straight line? Are you okay on what u_xx means as far as the shape of u(t,x)? And the shape I correlate strongly with the idea that spatial...
5. PDE and heat equation

Convexity. Heat flows in a region only if there is convexity in the spatial distribution of temp. Convexity is the second derivative, or in this case, convexity in the spatial direction is the second partial with respect to x.
6. PDE and heat equation

you're missing a time derivative, \partial_tu=\partial_{xx}u. We have the second partial of temperature in the spatial direction. One way to see this is, consider a local maximum in temperature in a spatial distribution of temp. Then all points nearby are warmer, so as time proceeds, the...

But isn't there a paradox in there? If they both accelerate away from each other, shouldn't they both be two years younger than each other. That looks like an apparent paradox to me.
8. Linear algebra proof

Oops, yeah you're right aren't you. I guess because the slick proof I was thinking of, was take x.e_1=x_1=0. That works right? So I was just guessing that it relied on some qualities of the basis vector, but maybe the real mistake would be to not refer to the fact that they are linearly...
9. Physical significance of gauge invariance

I'll let someone else cover this, but before they do, I'd like to point out that at least I'm confused on a few points. My uninformed guess, others will have a hard time responding to your last post. On the following points: How is a gauge translation relating A and B, AND symmetries of...
10. Find dy/dx if f(x,y) = 0

This reply doesn't really stand alone, I was trying to build on what others had previously said. If f is a nice function (for instance, a polynomial in x and y, or some other equation not doing anything crazy. Look up space filling curve.), the solution set is one-dimensional. So it is a curve...
11. Physical significance of gauge invariance

First guess: gauge invariance I associate with the idea of symmetries and conservation laws. So if some aspect of our universe cannot tell the difference for some transformation, or symmetry, we can call that a change in gauge. For instance, we can't tell if the universe shifts back one...
12. Linear algebra proof

The problem with the above proof, it doesnt seem to use the fact that the basis is orthonormal. You could potentially "prove something false".
13. Linear algebra proof

Think about what relations the basis vectors satisfy, if you notice the right thing, the proof is pretty swift.
14. Two questions about cycles (algebra)

In a nutshell, notice that alpha gamma alpha inverse takes alpha of 1 to alpha of 3. ;)

Because forces point downhill. In math, the purer idea is uphill.
16. Differential equation question

I would say it is similar to solving quadratic equations, saying ax^2+bx+c=0. We could move the c over, but after factoring to (x-x_1)(x-x_2)=0, it is easier to read off the roots. Also, sometimes there is a function of t on the right, y''-5y'+6y=f(t), and the business on the left is...
17. Coefficients derivative

My guess, the behavior of the solution set changes drastically wherever m(x)=0.
18. Deeper understanding of the gradient and directional derivative

One way to simplify the understanding, is to work all this out for the geometry of a linear function, z=cx+dy. If you move on unit along x, then one unit along y, you've traced the edge of a parallelogram situated above a unit square, and you've risen c+d units. There, that's why the total...
19. Quotient group

My favorite quotient groups is the numbers on a face clock. The numbers ...-11,1,13,25... are an equivalence class, with representative 1. Thus, the infinite set of integers ...-1,0,1,2,3,... is partitioned into 12 sets, or cosets. Since the theorem of quotient groups holds, these 12...
20. What is the significance of compactness?

Your definition of compact is rather abstract, and in fact, it is the definition of compact for general topologcial spaces, not just in the topci of metric spaces you are studying here. In metric spaces, this defnition, if i recall, is equivalent to closed and bounded. In earlier...
21. A discrete subset of a metric space is open and closed

No sweat, best of luck. Yeah, having to rush through math is not fun, it's sort of like the squirrel who saves nuts for the winter is well off, school is much more of a pain when we're rushing. But I bet you're enjoying it, and don't let my impulsive crankiness stop you from thinking out...
22. A discrete subset of a metric space is open and closed

You are wrong, find the mistake. Hmm. You need to slow down and specify what X is. You seemed to have two different claims before you started your proof, so I must confess I am not going to take the time to read this proof, since I don't know what your claim is. This is not true in...
23. A discrete subset of a metric space is open and closed

How would you prove this?
24. Integration of complex funtions

I like to see the subject of complex analysis as having two applications (at least that you can learn about fairly early on, there's more, but then I'd have to charge you (JK, I don't know more, it's above my paygrade)). One is, it helps to do some difficult one-d integrals, the subject of...
25. Maximizing problem with an inequality constraint.

In case (iii), x_1=0. Why not just go back to the original conditions, you are trying to maximize y=2*x_2 subject to the constraint x_2<=10, so clearly we have a max when x_2=10, thus y=20.
26. A discrete subset of a metric space is open and closed

So you are claiming that isolated points cannot be interior? How would you prove this?
27. A discrete subset of a metric space is open and closed

I think you've decided E is closed. I agree. I can't confirm your proof however, as I can't stand trying to remember or look up all those derived set, limit points, adherent point defintions. You've got a great attitude, don't worry about being too exact on a forum, I'm being a crank. I'm...
28. Vector Calculus Question about Surface Integrals

What do you mean by polar coordinates? Did you mean spherical, cylindrical?
29. What are Divergence and Curl?

divergence and curl are like two types of derivatives for vector fields. sort of like how cross product and dot product are two types of multiplication for vectors. best to imagine our vector field for a water flow; at each point, the vector is measuring the speed and direction of the flow...
30. A discrete subset of a metric space is open and closed

I think you mean p \in E, not p \subset E. Sounds good. Not sure how you got that. I think you mean E is not open. That does not imply that E is closed. This misunderstanding I've heard of before. In defintions, like that of open, the "if" really means "if and only if". But defintions will...