Problem I'm having with intrinsic equations when studing differential geometry

In summary, the conversation discusses the concept of intrinsic equations and its application to the given curve y=ln|secx|. The discussion includes a question on proving that \frac{ds}{dx} = secx and finding the intrinsic equation. The answer provided is incomplete but shows the understanding of the principle behind intrinsic equations and the attempt at solving the question. The conversation ends with a request for clarification and further help on completing the question.
  • #1
finchie_88
I've just started learning about intrinsic equations after learning about envelopes, arc length and curved surface area in cartesian, parametric and polar coordinates, and understand the principle behind intrinsic equations, but my book doesn't explain it particularly well, here is a question I'm stuck on and my attempt so far:

Question:
For the curve [tex] y= \ln|secx| [/tex] prove that [tex]\frac{ds}{dx} = secx [/tex] and that [tex]\psi = x [/tex]. Hence find the intrinsic equation.

My answer (Very incomplete):
[tex]y = \ln|secx| \therefore \frac{dy}{dx} = tanx[/tex]
[tex] \frac{ds}{dx} = \frac{ds}{dy} x \frac{dy}{dx} [/tex]
[tex] \frac{ds}{dx} = \sqrt{1 + (\frac{dx}{dy})^2} = \frac{tanx.secx}{tanx} = secx [/tex]

I don't know how to prove that [tex] x = \psi [/tex], but I think for the last part that [tex] s = \int sec \psi . d \psi [/tex], is this correct?

Can someone explain or show me if what I have so far is right, and how to complete the question? any help would be appreciated.

edit: Where it says ds/ds = ds/dy x dy/dx, the x in the middle is suppose to be a multiply sign.
 
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  • #2
Doesn't matter, after a lot of thought and a little scribbling I realized how to prove that [tex] \psi=x [/tex], and that my answer was correct.
 

1. What are intrinsic equations in differential geometry?

Intrinsic equations in differential geometry refer to equations that are independent of the coordinate system used. They describe geometric properties of a mathematical object, such as curvature or distance, without relying on any specific coordinate system.

2. Why are intrinsic equations important in differential geometry?

Intrinsic equations are important because they allow us to study and understand geometric properties of an object without being limited by a specific coordinate system. This allows for a more general and abstract understanding of the object, which can lead to deeper insights and applications.

3. How do intrinsic equations differ from extrinsic equations?

Intrinsic equations describe geometric properties of an object from within the object itself, without any reference to an external space. Extrinsic equations, on the other hand, describe the position and orientation of an object in relation to an external space.

4. What are some common applications of intrinsic equations in differential geometry?

Intrinsic equations are used in a wide range of fields, including physics, engineering, and computer graphics. They are particularly useful in understanding and describing the curvature of space, such as in the theory of relativity, and in developing algorithms for computer simulations.

5. How can I improve my understanding and use of intrinsic equations in differential geometry?

To improve your understanding and use of intrinsic equations, it is important to have a strong foundation in differential geometry and its mathematical concepts. Practice using intrinsic equations in various applications and contexts, and seek out resources such as textbooks, online courses, and lectures to deepen your understanding.

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