Homework Answers
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x +...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
4. Let f(x, y) = (xy, r2 + y). Note that f(1, 2) = (2,5). (a)...
4. Let f(x, y) = (xy, r2 + y). Note that f(1, 2) = (2,5). (a) Show that has a smooth inverse f-1 in a neighborhood of the point (1,2). (b) Find the differential matrix D(-)(2,5).
Consider the following subsets of R2: C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0} C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0} Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union...
Consider the following subsets of R2:
C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0}
C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0}
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Let y: 1 + R2 be a regular parametrised curve which we write as y(t) =...
Let y: 1 + R2 be a regular parametrised curve which we write as y(t) = (v(t), v(t))" for some smooth maps u,v: 1 R. We assume furthermore that is never equal to zero on I. We define the surface of revolution Exy associated to y as (1) E = {r(t,0) = (v(t) cos(6), y(t) sin(0), v(0))?|tel, 0 € (0,27]} . Below, we consider the chart (U,r) obtained by taking U = I x (0,27), where the map r:U →...
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y)...
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y) Lateral surface xy) As C: Curve in xy-plane
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a)...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
Consider the solution to the IVP y' - xy = x; y(0) = 2 Find y'...
Consider the solution to the IVP y' - xy = x; y(0) = 2 Find y' (0) Consider the solution to the IVP y' - xy = t; y(0) = 2 Find y" (0)
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (...
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
Consider the equation 3x²y" + x(2 – xy + xy = 0 with regular singular point Xo = 0. (a) Find the indicial roots ri, r2, with ri r2. Show your calculations. (b) Which of the following is true for the equation above: Indicate the letter of your choice and explain your choice. % There are two linearly independent convergent series solutions of the form yı (x) = x Š cux" and y(x) = x Š b,x". H0 N=0 (1)...
4. Consider the functions f : R2 R2 and g R2 R2 given by f(x, y) (x, xy) and g(x, y)-(x2 + y, x + y) (a) Prove that f and g are differentiable everywhere. You may use the theorem you stated in (b) Call F-fog. Properly use the Chain Rule to prove that F is differentiable at the point question (1c). (1,1), and write F'(1, 1) as a Jacobian matrix.
4. Consider the functions f : R2 R2 and...
4. Let f(x, y) = (xy, r2 + y). Note that f(1, 2) = (2,5). (a) Show that has a smooth inverse f-1 in a neighborhood of the point (1,2). (b) Find the differential matrix D(-)(2,5).
Consider the following subsets of R2:
C1 ={(x,y)∈R2 :x+y≤3,x≥0,y≥0}
C2 ={(x,y)∈R2 :4x+y≤4,x≥0,y≥0}
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each set is convex or not. If the set is not convex, give an example of a line segment for which the definition of convexity breaks down
Algebra Consider the following subsets of R2. Draw a sketch of the intersection CinC2 and the union C1UC2. State whether each...
(6) Show that the semicircle C = {(x,y) = R2 | + y2 = 1, y > 0} is a 1-dimensional manifold with boundary and the hemisphere D= {(x, y, z) | 22 + y2 + z2 = 1, 2 > 0} is a 2-dimensional manifold with boundary. (7) Suppose X is an n-dimensional manifold with boundary. Let ax denote the set of points in the boundary of X. Show that ax is an (n-1)-dimensional manifold.
Let y: 1 + R2 be a regular parametrised curve which we write as y(t) = (v(t), v(t))" for some smooth maps u,v: 1 R. We assume furthermore that is never equal to zero on I. We define the surface of revolution Exy associated to y as (1) E = {r(t,0) = (v(t) cos(6), y(t) sin(0), v(0))?|tel, 0 € (0,27]} . Below, we consider the chart (U,r) obtained by taking U = I x (0,27), where the map r:U →...
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y) Lateral surface xy) As C: Curve in xy-plane
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface...
4. Consider two functions x(h),y (h), Vh> 0 and cross-covariance function 7xy(h), Vh e Z (a) (7 points) Find the Best Linear Predictor (BLP) of Y given Xt, and its Mean Square Prediction Error (MSPE), expressed in terms of x(h). (h), xy (h) jointly stationary time series (X, Y), with individual auto-covariance (b) (13 points) Find the BLP of Y given Xt, X-1,Y-1, and its MSPE, expressed in terms of 7x(h)(h), x.y (h). (Note: you don't need to solve the...
Consider the solution to the IVP y' - xy = x; y(0) = 2 Find y' (0) Consider the solution to the IVP y' - xy = t; y(0) = 2 Find y" (0)
Question 8 (15 marks) Consider the function f: R2 R2 given by 1 (, y)(0,0) f(r,y) (a) Consider the surface z f(x, y). (i Determine the level curves for the surface when z on the same diagram in the r-y plane. 1 and 2, Sketch the level curves (i) Determine the cross-sectional curves of the surface in the r-z plane and in the y- plane. Sketch the two cross-sectional curves (iii) Sketch the surface. (b) For the point (r, y)...
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