![xerc 0 e:0 y ln(r), Let Dbe the following two dimensional donain. D := {(zy) E R2 : 1 S$ where In denotes the natural logarithm and the umber e denotes its base, ie. e 2.71. (1) Sketch the domain D and compute its area. (2) Let us define the domain D as the part of the rectangle [I.ej x (o, I] that is above D. Sketch D and compute its area. (3) Compute Now rotate the domain D around the r-axes to obtain a solid 3D object which is axially symmetric w.r.t. the z-axes. Call this domain W (4) Sketch the domain W and compute its volume Hint: fnd out what are the cross sections of this object if you eut it with planes parallel to the y-axis. (5) Compute j j fwysdV(z, ) Hint: use the previously determined cross sections and use eventually polar coordinates in the ys-plane.](http://img.homeworklib.com/questions/83e440c0-d505-11ea-bbdf-755e51f3d025.png?x-oss-process=image/resize,w_560)
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xerc 0 e:0 y ln(r), Let Dbe the following two dimensional donain. D := {(zy) E...
Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the i...
Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation x=au + bu, y=cu + dv. (a) Sketch D in the r-y plane for the case ad -bc > 0. (a) Sketch D in the r-y plane for the case ad bc < 0. (c) Calculate the area of D. Show all working.
Let 0
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image...
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the x-y plane for the case ad - bc > 0. (a) Sketch D in the z-y plane for the case ad-bc 〈 0. (c) Calculate the area of D. Show all working.
(7) Let 0くa 〈 b 〈 c 〈...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Let 0 < a <b<e<d for a, b, c, d E R. Consider the set S={(u, ujo < u < 1, 0<u<1) and let D be t...
Let 0 < a <b<e<d for a, b, c, d E R. Consider the set S={(u, ujo < u < 1, 0<u<1) and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the r-y plane for the case ad - be>0. (a) Sketch D in the r-y plane for the case ad - be < 0. (c) Calculate the area of D. Show all working.
Exercise 6.B.3. Let the pair of random variables (X, Y) have joint density function f(x, y)-16(x-y)2...
Exercise 6.B.3. Let the pair of random variables (X, Y) have joint density function f(x, y)-16(x-y)2 įf x, y e [0, 11, 0 otherwise. a. Confirm that f is a joint density function by verifying that equation (6.B.4) holds, and use a computer or graphing calculator to sketch its graph. b. Compute the marginal density function of Y c. For each x e [0,1], compute the conditional density of Y given x. d. Compute the conditional expectation function E(Y|X =...
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+...
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and...
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0...
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0 else where S-((x, y, z) 1x2 + y2 +#51). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the...
Let 0< a<b<e<d for a, b, c, d E R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation x=au + bu, y=cu + dv. (a) Sketch D in the r-y plane for the case ad -bc > 0. (a) Sketch D in the r-y plane for the case ad bc < 0. (c) Calculate the area of D. Show all working.
Let 0
(7) Let 0くa 〈 b 〈 c 〈 d for a,b,c,d R. Consider the set and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the x-y plane for the case ad - bc > 0. (a) Sketch D in the z-y plane for the case ad-bc 〈 0. (c) Calculate the area of D. Show all working.
(7) Let 0くa 〈 b 〈 c 〈...
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
Problem 3 (12 points): Let D be a bounded domain in R" with smooth boundary. Suppose that K(x, y) is a Green's function for the Neumann . For each x E D, the function y H K(x, y) is a smooth harmonic For each x E D, the normal derivative of the function y K(x, y) . For each z e D, the function y K(x,y)-Г(z-y) is smooth near problem. This means the following: function on D(r satisfies (VyK(x, y).v(b))-arefor...
Let 0 < a <b<e<d for a, b, c, d E R. Consider the set S={(u, ujo < u < 1, 0<u<1) and let D be the region in the r-y plance that is the image of S under the variable transformation (a) Sketch D in the r-y plane for the case ad - be>0. (a) Sketch D in the r-y plane for the case ad - be < 0. (c) Calculate the area of D. Show all working.
Exercise 6.B.3. Let the pair of random variables (X, Y) have joint density function f(x, y)-16(x-y)2 įf x, y e [0, 11, 0 otherwise. a. Confirm that f is a joint density function by verifying that equation (6.B.4) holds, and use a computer or graphing calculator to sketch its graph. b. Compute the marginal density function of Y c. For each x e [0,1], compute the conditional density of Y given x. d. Compute the conditional expectation function E(Y|X =...
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
Problem 5. Letf: Z+Zbyn -n. Let D, E S Z denote the sets of odd and even integers, respectively. (a) Prove that fD CE, where D denotes the image of D under f. (b) Is it true that D = E? Prove or disprove. (c) Describe the set f[El. Problem 6. Letf: R R be the function defined by fx) = x2 + 2x + 1. (a) Prove that f is not injective. Find all pairs of real numbers T1,...
2. You are given the following multivariate PDF (x, y, z) E S X,Y,2(x, y,z)=)4m 0 else where S-((x, y, z) 1x2 + y2 +#51). (a) (5 points) Let T be the set of all points that lie inside the largest cylinder by volume that can be inscribed in the region of S. Similarly let U be the set of all points that lie inside the largest cube that can be inscribed in the region of S. What would the...
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