grandes-ecoles 2015 QII.B.3

grandes-ecoles · France · centrale-maths2__pc Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability)
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by $$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$ Let $U$ be the function defined by $$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$ Show that $T_U' = \delta_0$. 
If $T$ is a distribution on $\mathcal{D}$, we define the derivative distribution $T'$ by
$$\forall \varphi \in \mathcal{D}, \quad T'(\varphi) = -T(\varphi')$$
Let $U$ be the function defined by
$$\begin{cases} U(x) = 1 & \text{if } x \geqslant 0 \\ U(x) = 0 & \text{if } x < 0 \end{cases}$$
Show that $T_U' = \delta_0$.
Paper Questions
QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QII.B.5 QII.C.1 QII.C.2