what is the geometric meaning of this limit?

Question

Lim (dy-->0) of [f(x,y+dy)-f(x,y)]/dy

I am not sure what the geometric meaning of this limit is?
I think it represents a slope... but a slope of what ? a tangant? if so in what plane?

could some one please help me clearify the geometric meaning of this limit

Thank you

Answer 1

It's the slope of the surface f(x,y), in the y direction.

Since f() is a function of both x and y, it is often though of as the height of a surface above the x-y plane.

For example, I will make up some numbers: x = 2 and y = 6, then f(x,y) = 10. Now move to the point x =2, y = 6.001. You find that f(2, 6.001) = 10.003. The slope, in the y direction (the direction you moved in) is 0.003 / 0.001 = 3.

Answer 2

Yes, good thinking. This limit does represent the slope of a line tangent to a surface. I assume that you already know certain things about functions of two variables (such as: their graphs are surfaces in 3-D). If I've assumed too much, please ask again and I will clarify.

The upstairs terms are f(x,y+dy) and f(x,y). Notice how they have the same first coordinate: just x. Nothing happening there. However, in the second coordinate, there's a little bit of change going on. We start at the point (x,y) and head a small distance dy in the positive y direction. Adding dy to the y coordinate brings us to the point (x, y+dy).

In the xy plane, these two points lie on a vertical line (a line parallel to the y-axis). In 3-space (where this limit really lives), the points (x,y, f(x,y)) and (x,y+dy,f(x,y+dy)) lie in a plane parallel to the yz plane.

To cut to the chase: If you are a skier at the point (x,y) on the mountain called "f", the limit in question is the slope you see before you when you face north. Positive means up, and a lot of awkward walking; negative means down, so zoom away.

Answer 3

If you think of z=f(x,y) as a graph in xyz space then that expression represents slicing the graph with an xz plane to get a function of just y (at that fixed x) and getting the local change in z, ie f, versus y.

Somehow that description doesn't seem very clear, does it?

I think that you're on the right track in that it represents a tangent to the curve in the xz plane or the local df/dy

Answer 4

This is the partial derivative of the function z=f(x,y) with respect to y holding x constant.

If you take x=xo constant ( a plane paralel to the yz plane)
z=f(xo,y) becomes a function of y in this plane and the partial derivative with respect to y represents the slope of the tangent to the curve at point (xo,y,z)