Suppose x ˙ = f ( x ) is a smooth vector field on R 2 , plot the phase portrait for the system r ˙ = r ( 1 - r 2 ) [ r 2 sin 2 θ + ( r 2 cos 2 θ - 1 ) 2 ] and θ ˙ = r 2 sin 2 θ + ( r 2 cos 2 θ - 1 ) 2 . Sketch x vs. t for the trajectory starting away from the unit circle. Explain at t → ∞ . Concept Introduction: ➢ Perform the polar to Cartesian conversion and obtain the system equation in the Cartesian form. ➢ Plot the system equation. ➢ Plot the x vs. t plot.

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Answer 1

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 7.3, Problem 11E

Interpretation Introduction

Interpretation:

Suppose x˙ = f(x) is a smooth vector field on R2, plot the phase portrait for the system r˙ = r(1 - r2)[r2sin2θ + (r2cos2θ - 1)2] and θ˙ = r2sin2θ + (r2cos2θ - 1)2. Sketch x vs. t for the trajectory starting away from the unit circle. Explain at t→∞.

Concept Introduction:

➢ Perform the polar to Cartesian conversion and obtain the system equation in the Cartesian form.

➢ Plot the system equation.

➢ Plot the x vs. t plot.

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Answer 2

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 7.3, Problem 11E

Interpretation Introduction

Interpretation:

Suppose x˙ = f(x) is a smooth vector field on R2, plot the phase portrait for the system r˙ = r(1 - r2)[r2sin2θ + (r2cos2θ - 1)2] and θ˙ = r2sin2θ + (r2cos2θ - 1)2. Sketch x vs. t for the trajectory starting away from the unit circle. Explain at t→∞.

Concept Introduction:

➢ Perform the polar to Cartesian conversion and obtain the system equation in the Cartesian form.

➢ Plot the system equation.

➢ Plot the x vs. t plot.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 3

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 7.3, Problem 11E

Interpretation Introduction

Interpretation:

Suppose x˙ = f(x) is a smooth vector field on R2, plot the phase portrait for the system r˙ = r(1 - r2)[r2sin2θ + (r2cos2θ - 1)2] and θ˙ = r2sin2θ + (r2cos2θ - 1)2. Sketch x vs. t for the trajectory starting away from the unit circle. Explain at t→∞.

Concept Introduction:

➢ Perform the polar to Cartesian conversion and obtain the system equation in the Cartesian form.

➢ Plot the system equation.

➢ Plot the x vs. t plot.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 4

Question

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

Chapter 7.3, Problem 11E

Interpretation Introduction