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...
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...
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...
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...
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.
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.
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...
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...
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...
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...