![1) (a) Show that the shortest path between two given points in a plane is a straight line, using plane olar coordinates (b) Let the path between two points lie in a 3-D space, and let the coordinates be parameterized by a time t, so that x-x(t). У-y (t), and z-z(t). Write the integral to be minimized and the Euler-Lagrange equations. [Hint: It may help, but its not necessary, to write all of the equations in vector form]. Find the extremal solution, and the resulting distance using this formulation.](http://img.homeworklib.com/questions/b0448cd0-77ef-11ea-8370-354516a6e4e1.png?x-oss-process=image/resize,w_560)
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1) (a) Show that the shortest path between two given points in a plane is a...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n ...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n a/r2, where r is the distance from the origin. Use Fermat's principle, that the integral (6.3) is stationary, to find the path of a ray of light travelling in a plane containing the origin. [Hint: Use two- dimensional polar coordinates and write the path as φ = φ(r). The Fermat integral should have the form .ff(ф, ф'.r) dr, where f(d, ф. r) is actually...
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the...
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b
3. Find all critical points of dt dt with...
2. Path difference between two points on an EM wave is 10pm. If the wavelength A=900...
2. Path difference between two points on an EM wave is 10pm. If the wavelength A=900 nm what is he phase difference? 3. Path difference between two EM wave is 1.2 um and the phase difference is T. What is A? 1. An EM wave is propagating in free space. A=900 nm and wavevector k has two components, x & z vhere. If E-field is oscillating along y direction write the exponential form of the EM wave equation. E0=150V/m, 4o...
Let the curve C in the (x, y)-plane be given by the parametric equations x =...
Let the curve C in the (x, y)-plane be given by the parametric equations x = e + 2, y = e2-1, tER. (a) Show that the point (3,0) belongs to the curve C. To which value of the parameter t does the point (3,0) correspond? (b) Find an expression for dy (dy/dt) without eliminating the parameter t, i.e., using de = (da/dt) (c) Using your result from part (b), find the value of at the point (3,0). (d) From...
АЗ. You are given that the plane P contains both the point and the line Ls, where Q has position ...
АЗ. You are given that the plane P contains both the point and the line Ls, where Q has position vec- tor q = i + 3, and L3 is given by the equation r = (0, i, 2) + λ(1, 3,-1) (where λ is a real parameter). i) Write down two vectors representing two different directions which lie in the plane P. [2 marks i) By using the cross product or otherwise, find a direction perpendicular to the plane....
Use Green's theorem in the plane to evaluate 4 K= anti-clockwise around the closed path C given by the curves: x-0,...
Use Green's theorem in the plane to evaluate 4 K= anti-clockwise around the closed path C given by the curves: x-0, -1 2 y 2 -2 r 2, -TT/2 <0< T/2, x = 0, 2 2 yz 1, r= 1, TT/2 2 0 2 -T/2 Evaluate the line integral ass a double integral using polar coordinates. Your answer should consist of a single number accurate to five decimal digits or as an exact rational expression. For example: 10.13906368 OR rounded...
Write equations of a tangent and a normal plane for the given curves at the given points: 9.8.1 is the question number....
Write equations of a tangent and a
normal plane for the given curves at the given points:
9.8.1 is the question number.
9.8.1.x-t - sint, y- cos t, z-4sint/2 at t-/2;
9.8.1.x-t - sint, y- cos t, z-4sint/2 at t-/2;
JUST ANSWER PART B A. A point mass m moves frictionlessly on a horizontal plane. An...
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
Q1: Here we consider finding the length of the shortest path between all pairs of nodes...
Q1: Here we consider finding the length of the shortest path between all pairs of nodes in an undirected, weighted graph G. For simplicity, assume that the n nodes are labeled 1; 2; : : : ; n, that the weight wij of any edge e = (i; j) is positive and that there is an edge between every pair of nodes. In this question, the goal is to solve this via dynamic programming. Note that the algorithm you will...
1. (20 points) Suppose B is the solid region inside the sphere 2+ y2 +2 4, above the plane = 1, a...
Please show all steps. Thank you, need to verify what I'm doing
wrong.
1. (20 points) Suppose B is the solid region inside the sphere 2+ y2 +2 4, above the plane = 1, and in the first octant (z, y, z 0)、z, y and z are measured in meters and the density over B is given by the function p(z, y, z)-(12 + y2 + ?)-1 kg/m3 (a) Set up and write the triple integral that gives the mass...
6.24Consider a medium in which the refractive index n is inversely proportional tor2: that is, n a/r2, where r is the distance from the origin. Use Fermat's principle, that the integral (6.3) is stationary, to find the path of a ray of light travelling in a plane containing the origin. [Hint: Use two- dimensional polar coordinates and write the path as φ = φ(r). The Fermat integral should have the form .ff(ф, ф'.r) dr, where f(d, ф. r) is actually...
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b
3. Find all critical points of dt dt with...
2. Path difference between two points on an EM wave is 10pm. If the wavelength A=900 nm what is he phase difference? 3. Path difference between two EM wave is 1.2 um and the phase difference is T. What is A? 1. An EM wave is propagating in free space. A=900 nm and wavevector k has two components, x & z vhere. If E-field is oscillating along y direction write the exponential form of the EM wave equation. E0=150V/m, 4o...
Let the curve C in the (x, y)-plane be given by the parametric equations x = e + 2, y = e2-1, tER. (a) Show that the point (3,0) belongs to the curve C. To which value of the parameter t does the point (3,0) correspond? (b) Find an expression for dy (dy/dt) without eliminating the parameter t, i.e., using de = (da/dt) (c) Using your result from part (b), find the value of at the point (3,0). (d) From...
АЗ. You are given that the plane P contains both the point and the line Ls, where Q has position vec- tor q = i + 3, and L3 is given by the equation r = (0, i, 2) + λ(1, 3,-1) (where λ is a real parameter). i) Write down two vectors representing two different directions which lie in the plane P. [2 marks i) By using the cross product or otherwise, find a direction perpendicular to the plane....
Use Green's theorem in the plane to evaluate 4 K= anti-clockwise around the closed path C given by the curves: x-0, -1 2 y 2 -2 r 2, -TT/2 <0< T/2, x = 0, 2 2 yz 1, r= 1, TT/2 2 0 2 -T/2 Evaluate the line integral ass a double integral using polar coordinates. Your answer should consist of a single number accurate to five decimal digits or as an exact rational expression. For example: 10.13906368 OR rounded...
Write equations of a tangent and a
normal plane for the given curves at the given points:
9.8.1 is the question number.
9.8.1.x-t - sint, y- cos t, z-4sint/2 at t-/2;
9.8.1.x-t - sint, y- cos t, z-4sint/2 at t-/2;
JUST ANSWER PART B
A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length ro is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr =-c(r-ro)" where c is a positive constant. Using plane polar coordinates r and θ: (i) Write down the Lagrangian L(r, θ,0) and use Lagrange's method to find the equations of motion for the mass...
Please show all steps. Thank you, need to verify what I'm doing
wrong.
1. (20 points) Suppose B is the solid region inside the sphere 2+ y2 +2 4, above the plane = 1, and in the first octant (z, y, z 0)、z, y and z are measured in meters and the density over B is given by the function p(z, y, z)-(12 + y2 + ?)-1 kg/m3 (a) Set up and write the triple integral that gives the mass...
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