Derivative of a discontinuous function on $\mathbb{R^2}$

Question

We define the function $\phi(x,t)=1_{x<\frac{t}{2}}(x,t)$.

And I want to calculate the partial derivatives of $\phi$ ($\partial_t\phi$ and $\partial_x\phi$).

Answer 1

The gradient ($\nabla \phi = (\partial_x \phi,\partial_t \phi)$ is 0 at any point other than the line $x=t/2$. On this line, it doesn't exist in the classical sense, but the distributional gradient does exist. For a vector valued $f\in C_c^\infty$, by Divergence theorem,

$$ \langle \nabla \phi,f\rangle := -\int_{\mathbb R^2} \phi(x,t) \nabla\cdot f(x,t) \, dxdt = -\int_{x<t/2} \nabla\cdot f(x,t) \,dxdt = -\int_{x=t/2} f\cdot n \, dl\\ = \int_{-\infty}^\infty f(\tau /2,\tau)\cdot\binom{1}{-1/2} d\tau$$