I have a function f(u,v) of two variables

If I set , how do I show then that

?

My book states this but I was wondering what rule was used to get this

Thanks very much

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- Aug 18th 2010, 10:11 AMslevviobit rusty on partial derivatives
I have a function f(u,v) of two variables

If I set , how do I show then that

?

My book states this but I was wondering what rule was used to get this

Thanks very much - Aug 18th 2010, 10:18 AMAckbeet
Well, the general rule would be

To me, off-hand, I'm a bit puzzled why the trig functions aren't in the denominators. Are you sure this is the correct expression? - Aug 18th 2010, 10:26 AMslevvio
yeah its about a parabolic cylinder, whose bottom runs along the v direction. Theta is introduced as an angle between v and w to show that we can see what happens in all directions for all theta, except when theta = 0, i.e. what happens in the v-direction. So perhaps we are considering theta to be constant here. I don't know. thanks for reminding me of the chain rule

- Aug 18th 2010, 11:13 AMAckbeet
I would definitely say that is constant here. But what I can't get over is where the trig functions are in the expression you're trying to prove. You've got

, so I would expect

A similar calculation would go for the other. I can't explain why this is not the case. Maybe there's something simple I'm missing. Maybe Danny could weigh in? - Aug 18th 2010, 11:30 AMslevvio
I will write out a section: ''

f has a maximum or a minimum (depending upon the sign) in the u-direction, but we do not yet know what happens in the v-direction. The surface z = f(x,y) is, to second order, a parabolic cylinder (Fig2.2)

In fact we know what happens in every direction except the v-direction. For let . Then at the origin

and

Hence f has the same sort of behaviour in the w-direction as in the u-direction, provided only that theta is not 0. IF theta is 0, i.e. in the v-direction, the Taylor series for f reduces to....

"

In case this is relevant this is an examination of what happens when the Hessian is 0 of a two variable function and not all the 2nd partial derivatives are zero - Aug 20th 2010, 08:15 AMJester
I think what you're trying to do (please correct me if I'm wrong) is to establish the directional derivative.

If we start at the point, say and move in the direction of say then

.

If we define then

which, by definition is . Using the chain rule for functions of more than one variable

so

noting that I've used and instead of and .

I would, however, like to know your reference. - Aug 20th 2010, 08:22 AMslevvio
the book is Saunders: an introduction to catastrophe theory.

thanks fo the help, i think it is the directional derivative. you can work it too out by making a w' variable which is orthogonal to the w axis and then rearrange and doing partial derivatives