You are Here: Home >< Maths

P5 Fun with Intrinsic Coordinates! watch

1. [Edexcel- Heinemann P5]

Since this text book seems woefully short of examples to practise converting Intrinsic to Cartesian coordinates and vice versa, I thought I would make my own questions by doing the given examples backwards. So I decided to start with the first given example [p104 example 10] which takes y=cosh x and derives s=tanψ from it.

Derive the Cartesian equation of s=tanψ

[Had to type psi in a couple of places. The editor seems to be playing up]

ds/dψ = sec²ψ
ds= sec²ψ.dψ

dy/ds = sinψ
dy = sinψ.ds

So dy = sinψ.sec²ψ.dψ
y=∫sinψ.sec²ψ.dψ

Parts: v=sinψ =>dv/dψ=cosψ
du/dψ = sec²ψ =>u=tanψ

y=sinψ.tanψ -∫tanψ.cosψ dψ
y=sinψ.tanψ -∫sinψ dψ
y=sinψ.tanψ +cosψ
y=sin²ψ/cosψ + cos²ψ/cosψ
y=secψ

[I've just noticed that sinψ.sec²ψ = secψ.tanψ, so I've gone a long way round here and I could have just quoted the standard ∫secψ.tanψ dψ = secψ, rather than showing it! Never mind. Good exercise...]

s=tanψ = dy/dx
1/s = dx/dy
dx = (1/s)dy
∫1.dx =∫(1/s)dy

and dy/ds = sinψ
dy = sinψ ds

So ∫1.dx =∫(1/s).sinψ ds =∫(1/tan psi).sinψ ds
ds=sec²ψ.dψ
so x=∫(1/tan psi).sinψ.sec²ψ.dψ = ∫secψ dψ
x= ln|secψ + tan ψ|

tanψ = √(sec²ψ -1)
and since y=secψ
x= ln|y + √(y² -1)| [logarithmic form of arcosh]
so x= arcsosh y
y = coshx QED

I'm sure a lot of this goes the long way round, but it's interesting.
I might not do a lot of these...

Anyone got any other Intrinsic-Cartesian workouts that are not in the Heinemann text?

Aitch
2. I agree that the heinemann book isn't exactly great at explaining this topic. It's important to have the right method for the possible situations:

1: Converting intrinsic form [s=f(w)] to cartesian form [y=g(x)].
Differentiate to give (ds/dw)=f'(w) so ds=f'(w) dw

(dy/ds)=sinw
∫1 dy = ∫sinw ds
y=∫sinw.f'(w) dw
Integrate to give a relationship between y and w. Remember not to neglect the arbitrary constant.

(dx/ds)=cosw
∫1 dx = ∫cosw ds
x=∫cosw.f'(w) dw
Integrate to give a relationship between x and w. Remember not to neglect the arbitrary constant.

Eliminate the parameter 'w' to give the cartesian relationship between y and x. Typically you'll use a trigonometric identity or make w the subject in the two equations to eliminate w.

2: Converting cartesian form [y=f(x)] to intrinsic form [s=g(w)]
y=f(x) so (dy/dx)=f'(x)=tanw
s=∫[1+(dy/dx)^2]^0.5 dx with lower limit a(where arc length is measured from) and upper limit x.
s=∫[1+(f(x))^2]^0.5 dx with lower limit a and upper limit x.
Perform the integration to find the relation s=h(x) and eliminate x using (dy/dx)=tanw, to form a relation involving only s and w.
Tou can convert from x=f(y) or parametric equations x=f(t), y=g(t) in a similar manner using the appropriate arc length formula.

I'm not sure whether anyone will find that useful but it's already typed now, it helps my own revision anyway at least
3. (Original post by Gaz031)
I agree that the heinemann book isn't exactly great at explaining this topic. It's important to have the right method for the possible situations:

1: Converting intrinsic form [s=f(w)] to cartesian form [y=g(x)].
Differentiate to give (ds/dw)=f'(w) so ds=f'(w) dw

(dy/ds)=sinw
∫1 dy = ∫sinw ds
y=∫sinw.f'(w) dw
Integrate to give a relationship between y and w.

(dx/ds)=cosw
∫1 dx = ∫cosw ds
x=∫cosw.f'(w) dw
Integrate to give a relationship between x and w.

Eliminate the parameter 'w' to give the cartesian relationship between y and x. Typically you'll use a trigonometric identity or make w the subject in the two equations to eliminate w.
I have to acknowledge that one of my most useful pieces of revision material is a printout of one of your earlier posts on this topic!

Thanks again!

Aitch
4. Thanks, guys.

Muchly useful.

Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: June 13, 2005
The home of Results and Clearing

3,045

people online now

1,567,000

students helped last year

University open days

1. Sheffield Hallam University
Tue, 21 Aug '18
2. Bournemouth University
Wed, 22 Aug '18
3. University of Buckingham
Thu, 23 Aug '18
Poll
Useful resources

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

How to use LaTex

Writing equations the easy way

Study habits of A* students

Top tips from students who have already aced their exams

Chat with other maths applicants