PDEsFunctionalAnalysis Using Multi-Index Notation, given and Misplaced &\int_{U}u \cdot D^\alpha \varphi \space dx &= \int_{U}u \cdot \partial_{x_1}^{\alpha_1}\cdots\partial_{x_{n}}^{\alpha_{n}}\varphi \space dx \\ &=(-1)^{|\alpha|}\int_{U}\partial_{x_1}^{\alpha_1}\cdots\partial_{x_{n}}^{\alpha_{n}} u \cdot \varphi \space dx \\ &= (-1)^{|\alpha|}\int_{U} D^\alpha u \cdot\varphi \space dx \end{aligned}$$ by integration by parts where the boundary terms vanish due to compact support. The notion of derivative can be generalized to the case when $u \notin C^k(U)$ in the following sense: the $\alpha^\text{th}$-**weak derivative** of $u$ is defined when there exists $$D^\alpha u := v \in L_\text{loc}^1(U)$$such that the above holds for all $\varphi \in C_c^\infty(U)$. Note that "$D^\alpha u$" is pure notation for some function that is locally integrable and satisfies the above. For a [[Bochner Integrable]] function $u \in L^1([0,T]; X)$, we define the weak derivative: $$\frac{d}{dt} u := v \in L^1([0,T]; X)$$ whenever $$\int_0^T \varphi'(t) u(t) \space dt = - \int_0^T \varphi(t) v(t) \space dt$$ holds for all $\varphi \in C_c^\infty([0,T])$.