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Math Help - Need to demonstrate that the slope is strictly positive

  1. #1
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    Need to demonstrate that the slope is strictly positive

    If I have a multivariable equation whose slope is always positive, say 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.


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  2. #2
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    Quote Originally Posted by rainer View Post
    If I have a multivariable equation whose slope is always positive, say 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.


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

    Tonio
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  3. #3
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    Oh yeah, good point.

    Let me give a few more parameters.

    First, I reduce the equation to 3 variables:

    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?
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  4. #4
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    I don't understand your definition. The definiton I am familiar with says a function f:\mathbb{R}^n\to \mathbb{R} 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 c^i are increasing, the derivative of f \circ c:\mathbb{R}\to \mathbb{R} is positive. Is this what you want?
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  5. #5
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    Quote Originally Posted by maddas View Post
    I don't understand your definition. The definiton I am familiar with says a function f:\mathbb{R}^n\to \mathbb{R} 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 c^i are increasing, the derivative of 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
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  6. #6
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    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?
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