Is $x=\int \int \ddot{x}\mathrm dx \mathrm dx$?
I know that multivariable calculus used like this
$$\phi = \int \int x \mathrm dx \mathrm dy$$
But, I was thinking to use double integral for single variable (double integral respect to single variable)
$$x=\int \int \ddot{x}\mathrm dx \mathrm dx$$
Is it correct? Or, there's better way to write it?
1 answer
While you can integrate twice for the same variable, your equation is not right. $\ddot x$ means deriving $x$ twice with respect to time, that is, $$\ddot x = \frac{\mathrm d^2x}{\mathrm dt^2}$$ which is very different from deriving twice with respect to $x$ (indeed, $\mathrm d^2x/dx^2=0$). To counteract time differentiation, you have to employ time integration, that is, $$\int\int \ddot x\,\mathrm dt\,\mathrm dt$$
Also, since you're using indefinite integrals, you have to take into account the constants of integration. That is, $$\int\int\ddot x\,\mathrm dt\,\mathrm dt = \int (\dot x + C_1)\,\mathrm dt = x + C_1 t + C_2$$
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