Sketch the phase portraits for the following systems. (Here r and θ are polar coordinates in the ...
Sketch the phase portraits for the following
systems. (Here r and θ are polar coordinates in the plane.)
a.
r' = r3 - 4r
θ' = 1
b.
r' = r(1 - r2)(9 - r2)
θ' = 1
c.
r' = r(1 - r2)(4 - r2)
θ' = 2 - r2
Homework Answers
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Sketch the phase portraits for the following systems. (Here r and θ are polar coordinates in the ...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f,...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
Problem 2. (a) Sketch the curves expressed here in polar coordinates r1=1+sin(0); r2 = 2 –...
Problem 2. (a) Sketch the curves expressed here in polar coordinates r1=1+sin(0); r2 = 2 – sin(6). (b) Find the area of the plane region that lies inside both curves: r1=1+sin() and r2 = 2 - sin(0)
5.6 In polar coordinates for R2, define the domain Ω={(r, θ): 0
Question on Partial Differential Equation
5.6 In polar coordinates for R2, define the domain Ω={(r, θ): 0
A curve in polar coordinates is given by: r = 9 + 2 cos θ Point P is at θ = 20π/18
A curve in polar coordinates is given by: r = 9 + 2 cos θ Point P is at θ = 20π/18 (1) Find polar coordinate r for P, with r > 0 and π < θ < 3π/2. (2) Find cartesian coordinates for point P (3) How may times does the curve pass through the origin when 0 < θ < 2π?
The Cartesian coordinates of a point are given. (2, −5) (i) Find polar coordinates (r, θ)...
The Cartesian coordinates of a point are given. (2, −5) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, wh...
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, where r < 0 and 0 < θ < 2π. Y= (b) The Cartesian coordinates of a point are -2,3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0)...
2. a) Show that the (signed) curvature for a curve in polar coordinates (r, 0) is given by where ...
2. a) Show that the (signed) curvature for a curve in polar coordinates (r, 0) is given by where ro denotes do Hint: derive the formulas r-r(0)cosa, y-r(θ)sin θ with respect to θ b) Compute the signed curvature for the cardioid r(0) 1-sin θ Sketch the curve with a suitable plotting tool.
2. a) Show that the (signed) curvature for a curve in polar coordinates (r, 0) is given by where ro denotes do Hint: derive the formulas r-r(0)cosa, y-r(θ)sin...
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8...
Acceleration in polar coordinates is required
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8)--, expressed in plane polar coordinates. The angle 6(t) changes with time according to the equation θ wt. Here a, are positive constants independent of time. (a) [10 marks) Compute the transverse acceleration of the particle (b) [10 marks) Find the force acting on a particle and express it in terms...
The polar coordinates of a certain point are (r = 3.50 cm, θ = 211°). The...
The polar coordinates of a certain point are (r = 3.50
cm, θ = 211°).
The polar coordinates of a certain point are (r = 3.50 cm, e = 211°). (a) Find its Cartesian coordinates x and y. x = -3.04 cm y = -1.8 cm (b) Find the polar coordinates of the points with Cartesian coordinates (-x, y). r = 3.53 cm e = -1.69 Your response differs significantly from the correct answer. Rework your solution from the beginning...
1. (This is problem 5 from the second assignment sheet, reprinted here.) Consider the nonlinear system a. Sketch the ulllines and indicate in your sketch the direction of the vector field in each of the regions b. Linearize the system around the equilibrium point, and use your result to classify the type of the c. Use the information from parts a and b to sketch the phase portrait of the system. 2. Sketch the phase portraits for the following systems...
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
(2) Let x-r cos θ, y-r sin θ represent the polar coordinates function f(r, θ) : R. R2, Compute f, (r$) and f, ( ompute * T
Problem 2. (a) Sketch the curves expressed here in polar coordinates r1=1+sin(0); r2 = 2 – sin(6). (b) Find the area of the plane region that lies inside both curves: r1=1+sin() and r2 = 2 - sin(0)
Question on Partial Differential Equation
5.6 In polar coordinates for R2, define the domain Ω={(r, θ): 0
(3 points) (a) The Cartesian coordinates of a point are (-1,-V3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0) of the point, where r < 0 and 0 < θ < 2π. Y= (b) The Cartesian coordinates of a point are -2,3) (i) Find polar coordinates (r,0) of the point, where r > 0 and 0 < θ < 2π. (ii) Find polar coordinates (r,0)...
2. a) Show that the (signed) curvature for a curve in polar coordinates (r, 0) is given by where ro denotes do Hint: derive the formulas r-r(0)cosa, y-r(θ)sin θ with respect to θ b) Compute the signed curvature for the cardioid r(0) 1-sin θ Sketch the curve with a suitable plotting tool.
2. a) Show that the (signed) curvature for a curve in polar coordinates (r, 0) is given by where ro denotes do Hint: derive the formulas r-r(0)cosa, y-r(θ)sin...
Acceleration in polar coordinates is required
1. A particle of unit mass moves along a trajectory , 2r) θ E (03), and θ E ( a coal, -a cose r(8)--, expressed in plane polar coordinates. The angle 6(t) changes with time according to the equation θ wt. Here a, are positive constants independent of time. (a) [10 marks) Compute the transverse acceleration of the particle (b) [10 marks) Find the force acting on a particle and express it in terms...
The polar coordinates of a certain point are (r = 3.50
cm, θ = 211°).
The polar coordinates of a certain point are (r = 3.50 cm, e = 211°). (a) Find its Cartesian coordinates x and y. x = -3.04 cm y = -1.8 cm (b) Find the polar coordinates of the points with Cartesian coordinates (-x, y). r = 3.53 cm e = -1.69 Your response differs significantly from the correct answer. Rework your solution from the beginning...
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