2. One of the many uses of vector fields in applied mathematics is to model fluid flow. Because we are confined to two dimensions in complex analysis, let us consider only planar flows of a fluid. This means that the movement of the fluid takes place in planes that are parallel to the ry-plane and that the motion and the physical traits of the fluid are identical in all planes. These assumptions allow us to analyze the flow of a single sheet of the fluid. Suppose that f(x) = P(x, y) + iQ(x, y) represents a velocity field of a planar flow in the complex plane. Then f(z) specifies the velocity of a particle of the fluid located at the point z in the plane. The modulus |f(2)|) is the speed of the particle and the vector f(z) gives the direction of the flow at that point. For a velocity field f(x) = P(r, y) + iQ(x, y) of a planar flow, the functions P and Q represent the components of the velocity in the x- and y-directions, respectively. If z(t) = x(t) + iy(t) is a parametrization of the path that a particle follows in the fluid flow, then the tangent vector z(t) = x(t) +iy(t) to the path must coincide with f(z(t)). Therefore, the real and imaginary parts of the tangent vector to the path of a particle in the fluid must satisfy the system of differential equations dx P(r, y) dt and dy Q(r, y) dt The family of solutions to the system of first-order differential is called the streamlines of the planar flow associated with f(z). Find the streamlines of the planar flow associated with f(z) = z?.
2. One of the many uses of vector fields in applied mathematics is to model fluid flow. Because we are confined to two dimensions in complex analysis, let us consider only planar flows of a fluid. This means that the movement of the fluid takes place in planes that are parallel to the ry-plane and that the motion and the physical traits of the fluid are identical in all planes. These assumptions allow us to analyze the flow of a single sheet of the fluid. Suppose that f(x) = P(x, y) + iQ(x, y) represents a velocity field of a planar flow in the complex plane. Then f(z) specifies the velocity of a particle of the fluid located at the point z in the plane. The modulus |f(2)|) is the speed of the particle and the vector f(z) gives the direction of the flow at that point. For a velocity field f(x) = P(r, y) + iQ(x, y) of a planar flow, the functions P and Q represent the components of the velocity in the x- and y-directions, respectively. If z(t) = x(t) + iy(t) is a parametrization of the path that a particle follows in the fluid flow, then the tangent vector z(t) = x(t) +iy(t) to the path must coincide with f(z(t)). Therefore, the real and imaginary parts of the tangent vector to the path of a particle in the fluid must satisfy the system of differential equations dx P(r, y) dt and dy Q(r, y) dt The family of solutions to the system of first-order differential is called the streamlines of the planar flow associated with f(z). Find the streamlines of the planar flow associated with f(z) = z?.
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Author:Raymond A. Serway, Chris Vuille
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Chapter1: Units, Trigonometry. And Vectors
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Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Transcribed Image Text:2. One of the many uses of vector fields in applied mathematics is to model
fluid flow. Because we are confined to two dimensions in complex analysis,
let us consider only planar flows of a fluid. This means that the movement of
the fluid takes place in planes that are parallel to the ry-plane and that the
motion and the physical traits of the fluid are identical in all planes. These
assumptions allow us to analyze the flow of a single sheet of the fluid. Suppose
that f(x) = P(x, y) + iQ(x, y) represents a velocity field of a planar flow in
the complex plane. Then f(z) specifies the velocity of a particle of the fluid
located at the point z in the plane. The modulus |f(z)|) is the speed of the
particle and the vector f(z) gives the direction of the flow at that point.
For a velocity field f(x) = P(x, y) + iQ(x,y) of a planar flow, the functions
P and Q represent the components of the velocity in the r- and y-directions,
respectively. If z(t) = x(t) + iy(t) is a parametrization of the path that a
particle follows in the fluid flow, then the tangent vector z(t) = x(t) +iy(t) to
the path must coincide with f(z(t)). Therefore, the real and imaginary parts
of the tangent vector to the path of a particle in the fluid must satisfy the
system of differential equations
dx
P(x, y)
dt
and
dy
Q(r, y)
dt
The family of solutions to the system of first-order differential is called the
streamlines of the planar flow associated with f(z).
Find the streamlines of the planar flow associated with f(z) = z'.
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