Partial Differentiation Notes

1, Suppose f(x,y) is a function. We say that lim f(x,y) = L when (x,y) approaches (a,b) if for every epsilon > 0 there is a delta > 0 so that whenever 0 < the square root of (x - a)^2 + (y - b)^2 < delta, |f(x,y) - L| < epsilon

2, f(x,y) is continuous at (a,b) if lim f(x,y) = f(a,b) with (x,y) approaching (a,b)

3, Polynomials are continuous everywhere. Rational functions are continuous everywhere they are defined

4, Let h = x - x0, t = y - y0, and e = z - z0 where z0 = f(x0,y0). The function z = f(x,y) is differentiable at (x0,y0) if

e = fx(x0,y0)h + fy(x0,y0)t + e1h + e2t

and both e1 and e2 approach 0 as (x,y) approaches (x0,y0)

5, If f(x,y) and its partial derivatives are continuous at a point (x0,y0), then f is differentiable there.

6, Suppose that z = f(x,y), f is differentable, x = g(t), and y = h(t). Assuming that the relevant derivatives exist,

dz/dt = dz/dx * dx/dt + dz/dy*dy/dt