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#1 2006-05-12 21:45:16

renjer
Member
Registered: 2006-04-29
Posts: 50

Proving an Equation when the function is unknown

Now here's a problem I have, it's often that I know the function of f but in this particular question, I do not know what f is at all. Is it okay if I let f be a function of my choice and prove it from there?

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#2 2006-05-12 22:01:57

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,966

Re: Proving an Equation when the function is unknown

renjer,
It is given w=f(x,y), x=rcosθ and y=rsinθ
and you are asked to show


You are right, w is given as a function of x and y, but the function is not defined sad


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2006-05-12 23:25:50

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: Proving an Equation when the function is unknown

IT IS A GOOD QUESTION!!!

It's a question about coords, combination of derivatives, and vector calculus altogether.

It's about expressing the original combination of derivatives in another co-ords, usually orthogonal curvy ones. For example, polar co-ords are curvy, and dr and dθ at any given point are orthogonal to each other.

It's usually explained in a vector calculus book or a calculus book for physicians. Untill now, I haven't got a satisfactory explaination for this kind of transfer in any books.

However, I do find a particular solution for your coords and your case.

Define

After substition,

According to Chain Rule


Thus you can compute the right of your equation and use cos²θ+sin²θ=1 to prove it equates the left.

Last edited by George,Y (2006-05-12 23:39:22)


X'(y-Xβ)=0

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#4 2006-05-13 01:33:32

renjer
Member
Registered: 2006-04-29
Posts: 50

Re: Proving an Equation when the function is unknown

I remember the chain rule telling me this:

So I'm quite unsure about your method from the chain rule onwards. Can you please explain further?

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#5 2006-05-13 01:59:16

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,966

Re: Proving an Equation when the function is unknown

renjer,
In this link, at the bottom of the page, you would find the chain rule for partial derivatives, which I think, George has referred.


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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