Definition of limit for f(x, y)
lim (x,y)→(x0,y0) f(x, y) = L if f(x, y) → L as (x, y) → (x0, y0) along every path.
Continuity of f(x, y) at (x0, y0)
f(x, y) is continuous at (x0, y0) if lim (x,y)→(x0,y0) f(x, y) = f(x0, y0). Continuous in region R if continuous at all points in R.
Definition of partial derivative with respect to x
fx = ∂f/∂x = lim h→0 [f(x + h, y) − f(x, y)] / h (y treated as constant).
Equality of mixed partial derivatives
If ∂f/∂x and ∂f/∂y are continuously differentiable, then: ∂²f/∂x∂y = ∂²f/∂y∂x.
Chain rule (one independent variable)
If y = y(x) and x = x(t), then: dy/dt = (dy/dx)(dx/dt).
Chain rule for z = f(x, y) with x = x(t), y = y(t)
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt).
Implicit differentiation (chain rule)
If f(x, y) = 0 defines y = y(x), then: dy/dx = − fx(x, y) / fy(x, y) = − (∂f/∂x) / (∂f/∂y), provided ∂f/∂y ≠ 0.
Chain rule for two independent variables
If z = f(x, y), with x = x(s, t), y = y(s, t): ∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t) ∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
