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Mathematics
Calculus
The partial derivative of a function that includes step functions
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[QUOTE="RPinPA, post: 6206394, member: 651116"] S is a function which takes on discrete values, and jumps instantaneously between those values. The time derivative is either 0 (between jumps) or doesn't exist (at jumps). Yes, in physics we do that. It's not exactly rigorous but makes a certain kind of sense. The Heaviside function has a vertical line in it. The slope of a vertical line is infinite. The usual approach to make a statement like the above semi-rigorous is to make a smooth approximation, for instance a function which makes the jump in finite time. And then take the limit as that finite time gets smaller and smaller. Sure, S is finite. But its rate of change, being a finite change divided by a time interval of 0, is infinite. Again, the slope of a vertical line is infinite. Sure. It's just a question of what approximation would be most convenient to work with. First thing that occurs to me is something like this: $$\hat I(R_j) = \begin{cases}0, & R_j \lt 7 \\ \frac{R_j-7}{a}, & 7 \leq R_j \lt 7 + a \\ 1, & R_j \geq 7+a \end{cases}$$ That interpolates from 0 to 1 over an interval of width ##a##, with a line of slope ##1/a##. This makes ##\hat I## continuous (I'm using a "hat" to indicate that this is a modified version of your indicator function). But its derivative is still not continuous. ##\hat I## is not differentiable at ##7## or ##7+a##. So my next thing to try for the interpolation is a cubic, a function of the form ##f(R-7) = a_0 + a_1(R-7) + a_2(R-7)^2 + a_3(R-7)^3## with the conditions that it is 0 and 1 at the two endpoints and has derivative 0 at those two endpoints. With a little bit of algebra that leads me to this: $$\hat I(R_j) = \begin{cases}0, & R_j \lt 7 \\ \frac{3}{a^2}(R_j - 7)^2 - \frac{2}{a^3}(R_j-7)^3, & 7 \leq R_j \lt 7 + a \\ 1, & R_j \geq 7+a \end{cases}$$ This version goes smoothly from 0 to 1 over the interval and is differentiable everywhere. But the piecewise nature might still cause you some headaches. So the other suggestion I have is to use something like ##\hat I(R_j) = \frac{1}{2} \left [1 + \tanh \left ( \frac {R_j - 7}{a} \right ) \right ]##, which becomes the step function in the limit. In all cases, you would take the limit as ##a \rightarrow 0##. [/QUOTE]
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The partial derivative of a function that includes step functions
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