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
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
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?
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?
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