Answer to Question #162940 in Classical Mechanics for fari

Question

 a) Explain the equation 𝑭 = −𝜵𝑈

Answer 1

$dA=\vec{F}d\vec{r}=-dU,$

$\vec{F}=F_x\vec{i}+F_y\vec{j}+F_z\vec{k},$

$d\vec{r}=dx\vec{i}+dy\vec{j}+dz\vec{k},$

$\vec{F}d\vec{r}=F_xdx+F_ydy+F_zdz=-dU,$

$F_x\partial x=-\partial U,~$ $F_x=-\frac{\partial U}{\partial x},$

$F_y=\frac{\partial U}{\partial y},$ $F_z=\frac{\partial U}{\partial z},$

$\vec{F}=-(\frac{\partial U}{\partial x}\vec{i}+\frac{\partial U}{\partial y}\vec{j}+\frac{\partial U}{\partial z}\vec{k}),$

$\vec{F}=-\text{grad}U,$

$\vec{F}=-\nabla U.$