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Slope of 3-Dimensional Equations

Date: 11/12/97 at 21:31:35
From: j.keenan
Subject: Slope of 3-dimensional equations

Is it possible to find the slope of three-dimensional equations?

Date: 11/13/97 at 18:20:49
From: Doctor Tom
Subject: Re: Slope of 3-dimensional equations

I'm not positive what you mean by "three-dimensional equations,"
but I'll guess you mean a function of two variables:

   z = f(x,y).

When you "plot" this, it has to be done in 3-D. It is a surface over 
the x-y plane, and the height of the surface is the z-value.  

For example, if the function is:

   z = x^2*y (x squared times y),

the height (z-value) above the point (2, 3) is 12.

"Slope" in one dimension can be represented by one number, namely how 
fast the curve is going up as you move to the right. In two dimensions 
(3-D plot), you need 2 numbers to determine the slope. One indicates 
how quickly the surface rises as you increase x, and the other, how 
quickly the surface rises as you increase y. For "nice" functions, 
these two numbers are enough to determine the rate of rise moving in
any direction in the x-y plane.

Mathematicians usually don't think of a two-numbered slope, though.  
It's best to re-think what slope means in one dimension as the line 
that just touches the curve at the point in question. The slope of 
this line is the slope of the curve. In 2-D, instead of a line tangent 
to a curve, think of a plane tangent to the surface. That tangent
plane is what's usually worked with.

The formula for the tangent plane at (x0,y0) is this:

   z = f(x0,y0) + x*f_x(x0,y0) + y*f_y(x0,y0)

where by f_x(x0,y0), I mean the partial derivative with respect to x, 
evaluated at the point (x0,y0).  f_y is the same, but with the partial 
derivative taken with respect to y.

This works in any number of dimensions, although beyond two, you can't 
plot it in our three-dimensional space. You need to have a good 
imagination of what 4D (or 5D, or 6D ...) space is like.

But the formulas are easy.  Here's what the "tangent space" to a 4D 
curve w = f(x,y,z) would look like at (x0,y0,z0):

w = f(x0,y0,z0) + x*f_x(x0,y0,z0) + y*f_y(x0,y0,z0) + z*f_z(x0,y0,z0)

This part of the expression above:

    x*f_x(x0,y0,z0) + y*f_y(x0,y0,z0) + z*f_z(x0,y0,z0)

is called the "differential".

-Doctor Tom,  The Math Forum
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Associated Topics:
College Calculus
High School Calculus

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