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Weak Derivative: Simple Definition, Examples

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The weak derivative uxi is an extension of the ordinary derivative to allow for the inclusion of problematic “points” that would otherwise make useful functions non-differentiable.

For example, the absolute value function cannot be differentiated on the interval [-2, 2] because of a sharp corner:


However, we can approach this problem from the backend: the function can be integrated (using integration by parts) [1]. We can then work backwards, creating a derivative that isn’t quite as “strong” as one we would have had if we could have used the usual routes for finding derivatives:
weak derivative
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We could choose to include the function’s corner value at x = 0 in either interval. The choice is arbitrary and makes no difference [2]. Obviously though, this weak derivative is forced to work on a space (i.e. the problematic corner) that it shouldn’t be defined on in the classical sense. We call this forced construct a Sobolev space, named after S. L. Sobolev, who first introduced the idea of a weak derivative [3].


We can perform this backwards magic because of the Fundamental Theorem of Calculus (FTC); Like their ordinary counterparts, weak derivatives satisfy the theorem. If the ordinary derivative exists, then the weak and ordinary derivatives are equivalent.

Another Piecewise Example

The piecewise function shown in the following image has a problematic corner at x = 1, which means that the ordinary derivative cannot be found at that point:

Graph: Desmos.



However, if the sharp corner is ignored, we can define the weak derivative:


Again, the choice of which interval to put the function value at the problematic point in (the first, second, both, or none) is not relevant.

Use of the Weak Derivative

Weak derivatives are usually used as an intermediate step towards a solution. For example, when looking for solutions to partial differential equations, it’s often easier to ask for weak derivatives and then show the solution is differentiable in the ordinary sense [4].

References

Desmos graphing calculator.
[1] Jost J. (1998) Integration by Parts. Weak Derivatives. Sobolev Spaces. In: Postmodern Analysis. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03635-8_21
[2] Sobolev Spaces. Retrieved May 6, 2021 from: https://www.math.ucdavis.edu/~hunter/pdes/ch3.pdf
[3] Soboleva, S. L. (1979). [in Russian]. On the mixed boundary-value problems for Sobolev-type equations with variable coefficients. Tr. Semin.
[4] Weber, J. (2018). Introduction to Sobolev Spaces. Retrieved May 5, 2021 from: http://www.math.stonybrook.edu/~joa/PUBLICATIONS/SOBOLEV.pdf

CITE THIS AS:
Stephanie Glen. "Weak Derivative: Simple Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/weak-derivative-simple-definition-examples/
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