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A professor drew a velocity field on the board and he placed a line segment in the field at some time ##t_1## and then drew the stick at some new time ##t_2##. My question, let's say we're given a velocity field ##\vec{V} = v_x \hat{i} + v_y \hat{j}## where ##v_x## and ##v_y## are the magnitudes of velocity. Given a point in space ##(p,q)##, after a certain amount of time, how would we tell where the point would be?

Thinking about this, if all we had was ##\vec{V} = f(x,y) \hat{i} + g(x,y) \hat{j}## then to determine where ##p## would go, we would say ##\partial_t p = f(x,y) \implies \int_{p(0)}^{p(t)} \, dp = \int_0^t f(x,y) \, dt## but here I am stuck.

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# I "stick" in velocity field

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