The **weak derivative** u_{xi} 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:

.

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:

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

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