A sufficient condition for the existence of total differential at a point: the function has all partial derivatives in a neighborhood of the point and all partial derivatives are continuous at the point.

The necessary condition for the existence of total differential at a certain point: all directional derivatives exist at that point.

Necessary and sufficient conditions for the existence of total differential at a certain point: if there is a binary function u(x, y) that makes the left end of the equation M(x, y)dx+N(x, y)dy=0 fully differential, that is, M(x, y)dx+N(x, y)dy=du(x, y), it is called total differential. What are the necessary and sufficient conditions for fully differential equations? M/？ y=？ N/？ X. Now it is generally called reciprocal relation or Euler reciprocal relation.

If the function

Z = all increments of f (x, y) at (x, y)

δz = f(x+δx，y+δy)-f(x，y)

It can be expressed as δ z = aδ x+bδ y+o (ρ),

Where a and b are independent of Δ x and Δ y, but only related to x and y, and Δ approaches 0 (Δ = √ [(Δ x) 2+(Δ y) 2]), then the function z=f(x, y) is differentiable at point (x, y), and a Δ x+b Δ y is called the function z=f.

Refer to the above content: Baidu Encyclopedia-Total Differential