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Total Derivative


There are at least two meanings of the term "total derivative" in mathematics.

The first is as an alternate term for the convective derivative.

The second meaning of total derivative is the derivative with respect to t of the function y=f(t,u_1,...,u_m) that depends on the variable t not only directly but also via the intermediate variables u_1=u_1(t,u_1,...,u_m),...,u_m=u_m(t,u_1,...,u_m). It can be calculated using the formula

 (partialy)/(partialt)=(partialf)/(partialt)+(partialf)/(partialu_1)(partialu_1)/(partialt)+...+(partialf)/(partialu_m)(partialu_m)/(partialt).

The total derivative of a function f(t,x,y,...) with respect to t is implemented in the Wolfram Language as Dt[f[t, x, y, ...], t].


See also

Convective Derivative, Exact Differential, Exterior Derivative

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References

Hazewinkel, M. (Managing Ed.). Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet "Mathematical Encyclopaedia." Dordrecht, Netherlands: Reidel, p. 228, 1988.

Referenced on Wolfram|Alpha

Total Derivative

Cite this as:

Weisstein, Eric W. "Total Derivative." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TotalDerivative.html

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