Differentiability is a property of a function where the limit of the rate of change of the function at a given point exists. In simpler terms, it means that the function is smooth and has a well-defined slope at that point.
The epsilon-delta method is a way to formally prove that a function is differentiable at a particular point. The basic steps involve setting an epsilon value (a small positive number), finding a delta value (a small positive number) that satisfies the given epsilon value, and then showing that the difference between the function values at two points is less than the given epsilon value when the distance between those points is less than the given delta value.
The process for proving differentiability of a multivariable function with the epsilon-delta method involves setting up the epsilon-delta definition of differentiability, finding a delta value that satisfies the given epsilon value, and showing that the difference between the function values at two points is less than the given epsilon value when the distance between those points is less than the given delta value.
To apply the epsilon-delta method to prove differentiability of f(x,y)=x^2-2xy, you would set up the epsilon-delta definition of differentiability, find a delta value that satisfies the given epsilon value, and show that the difference between the function values at two points is less than the given epsilon value when the distance between those points is less than the given delta value. This would involve using the partial derivative equations and the definition of a limit to find the appropriate delta value.
The key concepts to keep in mind when proving differentiability with the epsilon-delta method are the definition of differentiability, the epsilon-delta definition of differentiability, the use of partial derivatives, and the concept of a limit. It is also important to pay attention to the given epsilon value and ensure that the chosen delta value satisfies the given epsilon value.