Weak derivative Totally Explained

Everything about Weak Derivative totally explained

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, for example to lie in the Lebesgue space

L^1([a,b])

. See distributions for an even more general definition.

Definition

Let

u

be a function in the Lebesgue space

L^1([a,b])

. We say that

v

L^1([a,b])

is a weak derivative of

u

if, ť

int_a^b u(t)varphi'(t)dt=-int_a^b v(t)varphi(t)dt

for all continuously differentiable functions

varphi

with

varphi(a)=varphi(b)=0

.
Generalizing to

n

dimensions, if

u

and

v

are in the space

L_

This isn't the only weak derivative for u: any w that's equal to v almost everywhere is also a weak derivative for u. Usually, this isn't a problem, since in the theory of L^p spaces and Sobolev spaces, functions that are equal almost everywhere are identified.

Properties

If two functions are weak derivatives of the same function, they're equal except on a set with Lebesgue measure zero, for example, they're equal almost everywhere. If we consider equivalence classes of functions, where two functions are equivalent if they're equal almost everywhere, then the weak derivative is unique.
Also, if u is differentiable in the conventional sense then its weak derivative is identical (in the sense given above) to its conventional (strong) derivative. Thus the weak derivative is a generalization of the strong one. Furthermore, the classical rules for derivatives of sums and products of functions also hold for the weak derivative.

Extensions

This concept gives rise to the definition weak solutions in Sobolev spaces, which are useful for problems of differential equations and in functional analysis.

Further Information

Get more info on 'Weak Derivative'.

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