The Total Derivative As A Linear Map
Let be an open subset. Then a function is said to be (totally) differentiable at a point, if there exists a linear map (also denoted Dpf or Df(p)) such that
The linear map is called the (total) derivative or (total) differential of at . A function is (totally) differentiable if its total derivative exists at every point in its domain.
Note that f is differentiable if and only if each of its components is differentiable. For this it is necessary, but not sufficient, that the partial derivatives of each function fj exist. However, if these partial derivatives exist and are continuous, then f is differentiable and its differential at any point is the linear map determined by the Jacobian matrix of partial derivatives at that point.
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