/sci/ - Science & Math

>>11353898
You can treat them as infinitesimals, if you are careful. *In real life*, so long as you are dealing with first order, ordinary differentials, you can treat them as very small quantities awaiting a limiting process to zero and it doesn't matter. This is justified by how the first, total derivative is defined. Suppose you have a function [math]
x:t\to x(t) [/math]. Since we are in real life, you can safely assume that this function is infinitely differentiable and that it is continuous and all of its derivatives are continuous (this is where mathniggers want to lynch you). By definition
[eqn] \frac{\text{d} x}{\text{d} t}\equiv\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x}{\Delta t}=\lim_{\Delta t\to0}\frac{\Delta x}{\Delta t} [/eqn]
You can see that the derivative of x with resp. to t is dx/dt, which is defined as the limit of the ratio of two quantities. It is a property of limits that the limit of a ratio is the ratio of the limits and so it is totally legal to maneuver [math] \text{d}x\approx\Delta x [/math] and [math] \text{d}t\approx\Delta t [/math] around like algebraic quantities, cancelling and multiplying them and whatnot, so long as you keep in the back of your mind that this is all just shorthand. It makes math majors seethe, do it often. However, shit gets a lot more complicated when dealing with second derivatives or higher or partial derivatives, and the same logic definitely doesn't apply.
>>11353937
No, it isn't. Ductility, toughness, and hardness are also properties to consider. What would happen if a ball of glass collided with a ball of lead?