Differential geometry : Tangent vector & reparameterization

[Schwarzschild90]

Homework Statement

Problem statement uploaded as image.

Homework Equations

Arc-length function

The Attempt at a Solution

Tangent vector:
r=-sinh(t), cosh(t), 3

Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use the arc-length function to accomplish this.

 

[Ray Vickson]

Homework Statement

Problem statement uploaded as image.

Homework Equations

Arc-length function

The Attempt at a Solution

Tangent vector:
r=-sinh(t), cosh(t), 3

Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use the arc-length function to accomplish this.

What is preventing you from trying it for yourself?

 

[Schwarzschild90]

It's that I have no means of checking the solution, so before I invest in it, I would like to know if my method is correct (assuming that I integrate correctly).

 

[Mark44]

Homework Statement

Problem statement uploaded as image.

Homework Equations

Arc-length function

The Attempt at a Solution

Tangent vector:
r=-sinh(t), cosh(t), 3

This isn't the tangent vector.

Now, I just need to reparameterize it using arclength and verify my work is unit-speed. Will someone give me a hint? Should I use the arc-length function to accomplish this.

In future posts, please show more of your work. What you have here just barely qualifies as a problem attempt.

 

[Schwarzschild90]

Tangent vector

Now, compute the norm of the tangent vector:

Using this, make the following substitution

 

[Mark44]

Tangent vector
View attachment 93318

Now, compute the norm of the tangent vector:
View attachment 93319

What does "csgn" mean?

Using this, make the following substitution
View attachment 93320

What happened to the z-component of your tangent vector? I.e., the 3?

 

[Schwarzschild90]

How do I compute the arclength, without knowing the range? For example [0 <= t <= 2pi]

Another shot at the arc length of the tangent vector

$\sqrt{(9+9*sinh(t)^2+16*cosh(t)^2)}dt =^* 25 cosh^2(t) = 25 sinh^2(t)+25$

* Using a trigonometric identity

PS: csgn is code used specifically by maple. It' not necessarily a mathematical function

 

[Mark44]

How do I compute the arclength, without knowing the range? For example [0 <= t <= 2pi]

The arc length function in your relevant equations gives the arc length in terms of a parameter t.

Another shot at the arc length of the tangent vector

$\sqrt{(9+9*sinh(t)^2+16*cosh(t)^2)}dt =^* 25 cosh^2(t) = 25 sinh^2(t)+25$

The last expression above is not helpful, but the one before it is helpful. What happened to the square root?

* Using a trigonometric identity

PS: csgn is code used specifically by maple. It' not necessarily a mathematical function

Do you know what it means, though? I've never seen it, but I don't use Maple.

 

[Schwarzschild90]

Right, the square root should've been preserved, in the above equation. Here it is, in all of its glory:

$\sqrt{25cosh^2(t)}$
So, is this equation the reparameterization of the tangent vector?

csgn(x) is the sign function of real AND complex numbers; where csgn = complex signum.

 

[Ray Vickson]

Tangent vector
View attachment 93318

Now, compute the norm of the tangent vector:
View attachment 93319
Using this, make the following substitution
View attachment 93320

Why bother keeping the csign function? What does the graph of the cosh(t) function look like for real t?

 

[Schwarzschild90]

Plot of the 25cosh^2(t) function; the norm of the tangent vector

 

[Ray Vickson]

View attachment 93321
Plot of the 25cosh^2(t) function

This is not relevant. The question is what cosh(t) looks like, not its square. You should not even need to do an actual plot; just picture it in your mind.

 

[Schwarzschild90]

Plot of $5 \sqrt{cosh(t)}$

I can picture it in my mind. What am I supposed to "see"?

 

[Schwarzschild90]

I get this for the parameterization by arclength of the tangent vector
$int(5*sqrt(cosh(t)^2), t = 0 .. 1) = 5 sinh(1)$

 

[Mark44]

I get this for the parameterization by arclength of the tangent vector
$int(5*sqrt(cosh(t)^2), t = 0 .. 1) = 5 sinh(1)$

This is not a parameterization -- it's a number.

As I said before...

The arc length function in your relevant equations gives the arc length in terms of a parameter t.

IOW, $\int_0^t 5 \sqrt{\cosh^2(w)} dw$