Higher Derivatives

  • Kennan T. Smith
Part of the Undergraduate Texts in Mathematics book series (UTM)


The partial derivatives of a function f:RmRn are again functions from Rm to Rn which may have partial derivatives of their own. If so, the latter are called the second partial derivatives:
$$\frac{{{\partial ^2}f\left( a \right)}}{{\partial {x_i}\partial {x_j}}} = {D_{ij}}f\left( a \right) = {D_i}\left( {{D_j}f} \right)\left( a \right)$$

The formula says that to get D ij f(a), you take first Dif, which must exist on a neighborhood of a, and differentiate it with respect to xi. It appears that this would be quite different from D ji f (a), which is formed by differentiating first with respect to xi and then with respect to x j .In fact, the two are usually the same.


Manifold Dinates Teal 


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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Kennan T. Smith
    • 1
  1. 1.Mathematics DepartmentOregon State UniversityCorvallisUSA

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