Slope of 3-Dimensional EquationsDate: 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 Check out our web site! http://mathforum.org/dr.math/ |
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