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I) One mathematical problem is that the function $$\tag{1} f(q)~:=~ \frac{q\sin(rq)}{q^2+u^2}, \qquad q,r,u~>~0,$$ is not integrable $f\notin {\cal L}^{1}(\mathbb{R}_{+})$, because the integral over the absolute value of the integrand is infinite: $$\tag{2} \int_{\mathbb{R}_{+}} \! dq~|f(q)| ~=~\infty.$$ However it is still possible to define the ...

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The missing piece here is that the temperature of the resistor is a function of the current. Your equation should perhaps read $V = I\,R(T(I))$. Does that help?

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You have a function: $V(T, i) = i \cdot R(T)$ and you should get $\dfrac{dV}{di}$. $T$ doesn't change when you vary $i$ and $R(T)$ doesn't too, so it can be considered as a constant comparing to variable $i$. Fix $T$ at some generic value, for example $a$, doing this you get $R(T = a) = R_a$ So your function is reduced to $V(i) = i \cdot R_a$. Now you ...

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The other answers have related translation of a note into a frequency. For a guitar, I think that you wish to know the position of each point in the string as a function of time. Let the length of the string be L. Defined the point at which is plucked P along the length, with amplitude A. The initial position is a triangle (not necessarily isosceles) ...

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"How does one determine how quickly the string is oscillating between the two points in a unit of measurable distance?" The measure of how quickly the string is oscillating is its frequency, stated in Hertz (cycles per second). The frequency of played string is can be found by Ref: Piano key frequencies where n is the musical key number, e.g. middle C ...

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how fast is a stringed instrument's string interpolating? I assume you mean vibrating. Notes that are 1 octave apart vibrate at double the frequency (i.e. the note A should have the frequencies of 55, 110, 220, 440, 880, 1760, 3520, 7040 and 14080 Hz). The octave is divided by 12 to get our semitone scale. The frequency difference for each step is based ...

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Update: I guess I misunderstood the question. What I answered was "how fast do points on two strings move in relation to each other". Answer: The relative speed does not stay constant, but oscillates with the frequency equal to the difference in the frequencies. Sometimes the points on the string are in phase and the relative speed is zero but after a while ...

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Your statement about $|| r-r'||$ is true only if $r$ and $r'$ have the same $\phi$ coordinate. (same "longitude") The denominator does have a $\phi '$ dependance. The value of that modulus will be larger when $\phi \neq \phi '$.

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