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. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) 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. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).
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. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) 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. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).
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. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) 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. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).
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.
(a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines?
(b) 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.
(c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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