In the mathematical field of differential calculus, the term total derivative has a number of closely related meanings.
- The total derivative (full derivative) of a function, of several variables, e.g., etc., with respect to one of its input variables, e.g., is different from its partial derivative . Calculation of the total derivative of with respect to does not assume that the other arguments are constant while varies; instead, it allows the other arguments to depend on . The total derivative adds in these indirect dependencies to find the overall dependency of on . For example, the total derivative of with respect to is
- It refers to a differential operator such as
- It refers to the (total) differential df of a function, either in the traditional language of infinitesimals or the modern language of differential forms.
- A differential of the form
- It is another name for the derivative as a linear map, i.e., if f is a differentiable function from Rn to Rm, then the (total) derivative (or differential) of f at x∈Rn is the linear map from Rn to Rm whose matrix is the Jacobian matrix of f at x.
- It is a synonym for the gradient, which is essentially the derivative of a function from Rn to R.
- It is sometimes used as a synonym for the material derivative, in fluid mechanics.
Read more about Total Derivative: Differentiation With Indirect Dependencies, The Total Derivative Via Differentials, The Total Derivative As A Linear Map, Total Differential Equation, Application of The Total Differential To Error Estimation
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