Relativity: Exploring dX and dY in Math

[Stephanus]

No, not almost zero. $\lim_{h \to 0} \sin(h)$ IS zero.

I'm sorry Mark44.
if $lim_{h \to 0} \sin(h)$ is zero, how can

$lim_{h \rightarrow 0} \frac{sin(h)}{h} = 1$

I think $lim_{h \to 0} \sin(h) = h$. and also tan(h) = h

 

[Mark44]

I'm sorry Mark44.
if $lim_{h \to 0} \sin(h)$ is zero, how can

$lim_{h \rightarrow 0} \frac{sin(h)}{h} = 1$

As I said in a previous post, this limit is proved in many calculus textbooks.

I think $lim_{h \to 0} \sin(h) = h$. and also tan(h) = h

No, both of these are wrong. After you take a limit involving h, h will not appear in the result.
What is true is that for small h, $\sin(h) \approx h$ and $\tan(h) \approx h$, but both limits you showed above are zero.
$lim_{h \to 0} \sin(h) = 0$
and $lim_{h \to 0} \tan(h) = 0$ as well.

Before attempting to study advanced physics topics, you should get a calculus textbook and study it, or study it in an online course..

 

[micromass]

I think $lim_{h \to 0} \sin(h) = h$. and also tan(h) = h

This makes no sense since on the left side, $h$ is a dummy variable, while on the right it appears to be a real number.

 

[micromass]

Of course in 'math', the gradient for vertical length is undefined, but in SR a vertical line as you know, is a rest frame.
While in SR, the horizontal line is undefined, velocity is instant, but in 'math' a horizontal line is Y = n, the gradient is zero.

You probably don't want to use the word gradient here...

 

[jtbell]

You probably don't want to use the word gradient here...

To be more explicit:

In English-language math terminology, "gradient" is a concept from multivariable (vector) calculus, usually in three dimensions.

When we refer to the inclination of a line on a two-dimensional graph, e.g. a spacetime diagram for x and t in SR, we use the word "slope."

 

[Stephanus]

To be more explicit:

In English-language math terminology, "gradient" is a concept from multivariable (vector) calculus, usually in three dimensions.

When we refer to the inclination of a line on a two-dimensional graph, e.g. a spacetime diagram for x and t in SR, we use the word "slope."

Yeah, I"m an Indonesian and in high school time, we used the word "gradien", no "t" there
But that was 30 years ago. I don't know what the word now.
And in space time diagram for an accelerated world line, this "slope" is the velocity at that particular time?
Can I ask here.
In $Y = 4X + n$, the "slope" is "4"? Please confirm, so I can understand the explanations in SR Forum.
Thanks.

 

[jtbell]

And in space time diagram for an accelerated world line, this "slope" is the velocity at that particular time?

This is sneaky...

In classical (non-relativistic) physics we always draw these diagrams with t on the horizontal axis and x on the vertical axis. The slope is dx/dt which is indeed the velocity, v.

However, in relativistic physics, we customarily draw spacetime diagrams the other way around: x on the horizontal axis and t on the vertical axis. The slope is dt/dx which is the reciprocal of the velocity, 1/v. I don't remember why people started doing it this way. Someone in the relativity forum probably knows.