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Could someone solve the question and explain me

Let $\omega=(-1,1)\times (-1,1)$ and $f,g:\omega \to \mathbb{R}$ are defined by

$$f(x,y) = \begin{cases} 1, & \text{if $x$ $\le$ 0} \\[2ex] 0, & \text{anywhere else} \end{cases}$$ and

$$g(x,y) = \left\lVert x-y \right\rVert.$$

Determine whether $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$, $\frac{\partial g}{\partial x}$, $\frac{\partial g}{\partial y}$ exists weakly and state it if they exist.

Do I need to learn distributions to understand this ?

Juniven Acapulco
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AccGen
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  • Do you know the definition of weak derivatives? You do not need distributions to solve this (homework?) problem. – gerw Jan 12 '17 at 07:44
  • @gerw I do understand the definition. But what I do not understand is what norm should I apply for function g. I thinks it is square intergral norm. Is it right ? – AccGen Jan 12 '17 at 08:00
  • I do not know what you mean by "norm apply for function $g$". Why do you need a norm for the definition of weak derivatives? – gerw Jan 12 '17 at 08:09
  • @gerw the norm given in the question g(x,y) – AccGen Jan 12 '17 at 10:06
  • Oh, I see. Since $x, y$ are just scalars, I would suppose that this is the absolute value of the difference. Isn't it? – gerw Jan 12 '17 at 10:45
  • I found the derivative of a norm irrespective of the norm it is and here it is https://math.stackexchange.com/questions/291318/derivative-of-the-2-norm-of-a-multivariate-function – AccGen Jan 12 '17 at 18:17
  • This thread is only on the Euclidean norm! – gerw Jan 12 '17 at 18:31

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