Thread: Need to demonstrate that the slope is strictly positive

If I have a multivariable equation whose slope is always positive, say how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.


Thanks

Originally Posted by rainer

If I have a multivariable equation whose slope is always positive, say how do I demonstrate that the slope is always positive?

I imagine this involves partial derivatives but need some guidance.


Thanks

How do you define the slope of a multivariable function??

Tonio

Oh yeah, good point.

Let me give a few more parameters.

First, I reduce the equation to 3 variables:



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?

I don't understand your definition. The definiton I am familiar with says a function has positive slope iff for every injective curve a,b)\to \mathbb{R}^n" alt="c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" /> whose coordinate functions are increasing, the derivative of is positive. Is this what you want?

Originally Posted by maddas

I don't understand your definition. The definiton I am familiar with says a function has positive slope iff for every injective curve a,b)\to \mathbb{R}^n" alt="c=(c^1,\cdots,c^n)a,b)\to \mathbb{R}^n" /> whose coordinate functions are increasing, the derivative of is positive. Is this what you want?

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

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?