If I have a multivariable equation whose slope is always positive, say $\displaystyle f(x,y,z)=x^2+y^2+z^2$ how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.

Thanks

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- Feb 21st 2010, 08:37 AMrainerNeed to demonstrate that the slope is strictly positive
If I have a multivariable equation whose slope is always positive, say $\displaystyle f(x,y,z)=x^2+y^2+z^2$ how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.

Thanks - Feb 21st 2010, 11:10 AMtonio
- Feb 22nd 2010, 09:41 AMrainer
Oh yeah, good point.

Let me give a few more parameters.

First, I reduce the equation to 3 variables:

$\displaystyle f(x,y)=x^2+y^2$

So it's a 3D graph. I am interested in the slope on the 2D x-y plane or "cross-section" of the origin.

Does that narrow it down enough? - Feb 22nd 2010, 01:12 PMmaddas
I don't understand your definition. The definiton I am familiar with says a function $\displaystyle f:\mathbb{R}^n\to \mathbb{R}$ has positive slope iff for every injective curve $\displaystyle c=(c^1,\cdots,c^n):(a,b)\to \mathbb{R}^n$ whose coordinate functions $\displaystyle c^i$ are increasing, the derivative of $\displaystyle f \circ c:\mathbb{R}\to \mathbb{R}$ is positive. Is this what you want?

- Feb 24th 2010, 06:57 AMrainer

Hmmm, this definition is really cool-looking. But not understanding the half of it I will have to say I don't know if it's what I need or not.

It looks like I need to ruminate and clarify whatever it is I am trying to ask. So let's leave off here and maybe I'll post a clarified version of my question in a new thread.

Thanks a lot - Feb 24th 2010, 09:05 AMHallsofIvy
The difficulty appears to be that

**you**do not know what**you**mean by "slope" of a multivariable function. In order to be able to talk about slope being positive, you must mean it to be a number, but**what**number?