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  •  » Proving an Equation when the function is unknown

#1 2006-05-13 19:45:16

renjer
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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 2006-05-13 20:01:57

ganesh
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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


Character is who you are when no one is looking.

#3 2006-05-13 21:25:50

George,Y
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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 (2006-05-13 21:39:22)


X'(y-Xβ)=0

#4 2006-05-13 23:33:32

renjer
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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?

#5 2006-05-13 23:59:16

ganesh
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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.


Character is who you are when no one is looking.
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