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Question

# 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. If parametric equations of a flow line are $x = x(t), y = y(t)$ explain why these functions satisfy the differential equations $dx/dt = x$ and $dy/dt = -y$. Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).

Solution

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At any point $\dfrac{dx}{dt}$ is the $x$ component of the velocity.

At any point $\dfrac{dy}{dt}$ is the $y$ component of the velocity.

Here it is given that Velocity at the point $(x, y$) is $\textbf{F}(x, y) = x\textbf{ i }-y\textbf{ j }$

Note that, the $x$ component of the velocity is $x$ and

the $y$-component of the velocity is $-y$

Therefore, we can write

$\dfrac{dx}{dt} = x \rightarrow \textbf{(1)}$

$\dfrac{dy}{dt} = -y \rightarrow \textbf{(2)}$

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