The Flow Lines (Or Streamlines) of a Vector Field Are $\mathbf { F } ( x , y ) = 8 x \mathbf { i } - 32 y \mathbf { j }$

Short Answer

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. The flow lines of the vector field $\mathbf { F } ( x , y ) = 8 x \mathbf { i } - 32 y \mathbf { j }$ satisfy the differential equations $\frac { d x } { d t } = 8 x$ and $\frac { d y } { d t } = - 32 y .$ Solve these differential equations to find the equations of the family of flow lines.