Homework Answers
The solution is given below. I have simply used the definition of integrating differential forms over surfaces and then evaluated the double integral obtained.
(a) Let S be the area of a bounded and closed region D with boundary дD of a smooth and simple cl...
(a) Let S be the area of a bounded and closed region D with boundary дD of a smooth and simple closed curve, show that S Jlxy -ydx by Green's Theorem. (Hint: Let P--yandQ x) (b) Let D = {(x,y) 1} be an ellipse, compute the area of D a2 b2 (c) Let L be the upper half from point A(a, 0) to point B(-a, 0) along the elliptical boundary, compute line integral I(e* siny - my)dx + (e* cos...
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a 5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk...
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
Let W be the solid: 0 < 3,0 <y, 0 <z< 20 – 2y – X.,...
Let W be the solid: 0 < 3,0 <y, 0 <z< 20 – 2y – X., What is S?: S 20-2y 20-2y- S SSSw 1DV = S 1 dz do dy 0 0 0 Question 18 W is the same solid as in Question 17. What is T?: 20 T 20-2y-2 SSSw 1 DV = SS S 1 dz dy dx. 0 0 0 A) 10 B) 20 - 3 C C) 10 – D) 10 - y
Let W be the solid: 0 < x,0 <y, 0 <z < 20 – 2y -...
Let W be the solid: 0 < x,0 <y, 0 <z < 20 – 2y - X., What is S? S 20-2y 20-2y-2 SJSW 1DV = ſ s 1 dz dar dy S 0 0 Question 18 W is the same solid as in Question 17. What is T?: 20 T 20-2y-2 SSSw 1dV = SS S 1 dz dy dx. 0 0 0 A) 10 B) 20 C) 10 – C 2 D) 10 - y
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho,...
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho, do )-min( (k, d)such that k > 0, d 0, and the proces s W t ▽k▽dX,-(1 B)a Find ko and do. For W, (with k = ko and d...
Engineering Analysis Q.1. f(t) = {S; if - 4 <t<o if 0 st <4 a) Sketch...
Engineering Analysis
Q.1. f(t) = {S; if - 4 <t<o if 0 st <4 a) Sketch the function for 3 cycles [5 points ] b) Find the Fourier series for the function. [15 points)
2. Consider the set of curved coordinates t(t, s) in the plane R2(0,0) related to the Euclidian c...
2. Consider the set of curved coordinates t(t, s) in the plane R2(0,0) related to the Euclidian coordinates (r, y) by the transformations: 2 s2+ t . . t t ys , . (a) (10 points) Find Dx(t) := = (b) (5 points) Find the volume element dx dy expressed in the coordinates (s, t). Use that da dy detds dt 0(t,s) (c) bonus (10 points) Express the vector of first partial derivatives [, using the formula [a,,%) . via...
Sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(...
sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
Let E be the solid bounded by y+z=1 z=0 and y=x^2 a) Bind z, and provide (but do not evaluate) th...
Let E be the solid bounded by y+z=1 z=0 and y=x^2 a) Bind z, and provide (but do not evaluate) the triple integral with the plane described horizontally simple (dz dx dy) b) Bind z, and provide (but do not evaluate) the triple integral with the plane described vertically simple (dz dy dx) c) Bind x, and provide (but do not evaluate) the triple integral with the plane described horizontally simple (dx dy dz) d) Bind x, and provide (but...
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a)...
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a) Jls (cosz-s (b) fdl where y is the line segment from (2,-1,3) to (0, 1, 4) and f (x,y,z)-y+2 sin y) dA 3 marks 3 marks (c) Jc F dr where C is the unit circle centred at the origin, traversed once anticlockwise and F R2R2 is given by F(r,y)- (x2.x + y) 3 marks JJR eVEdA where R is the region enclosed by...
(a) Let S be the area of a bounded and closed region D with boundary дD of a smooth and simple closed curve, show that S Jlxy -ydx by Green's Theorem. (Hint: Let P--yandQ x) (b) Let D = {(x,y) 1} be an ellipse, compute the area of D a2 b2 (c) Let L be the upper half from point A(a, 0) to point B(-a, 0) along the elliptical boundary, compute line integral I(e* siny - my)dx + (e* cos...
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
5) Let Φ : R2-ל -(rcos(0), r sin(θ)), 0-r-R, 0-θ disk of radius R centered at (0,0)). Compute J dx Λ dy. R2 given by Φ(r, θ) -2n (this is a
Let W be the solid: 0 < 3,0 <y, 0 <z< 20 – 2y – X., What is S?: S 20-2y 20-2y- S SSSw 1DV = S 1 dz do dy 0 0 0 Question 18 W is the same solid as in Question 17. What is T?: 20 T 20-2y-2 SSSw 1 DV = SS S 1 dz dy dx. 0 0 0 A) 10 B) 20 - 3 C C) 10 – D) 10 - y
Let W be the solid: 0 < x,0 <y, 0 <z < 20 – 2y - X., What is S? S 20-2y 20-2y-2 SJSW 1DV = ſ s 1 dz dar dy S 0 0 Question 18 W is the same solid as in Question 17. What is T?: 20 T 20-2y-2 SSSw 1dV = SS S 1 dz dy dx. 0 0 0 A) 10 B) 20 C) 10 – C 2 D) 10 - y
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho, do )-min( (k, d)such that k > 0, d 0, and the proces s W t ▽k▽dX,-(1 B)a Find ko and do. For W, (with k = ko and d...
Engineering Analysis
Q.1. f(t) = {S; if - 4 <t<o if 0 st <4 a) Sketch the function for 3 cycles [5 points ] b) Find the Fourier series for the function. [15 points)
2. Consider the set of curved coordinates t(t, s) in the plane R2(0,0) related to the Euclidian coordinates (r, y) by the transformations: 2 s2+ t . . t t ys , . (a) (10 points) Find Dx(t) := = (b) (5 points) Find the volume element dx dy expressed in the coordinates (s, t). Use that da dy detds dt 0(t,s) (c) bonus (10 points) Express the vector of first partial derivatives [, using the formula [a,,%) . via...
sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
sin(s) cos(t)] Let S be the unit sphere, with the usual parameterization γ(st)-|sin(s)sin(t) cos(s) Let w zdz Λ dy. Find w.
Evaluate the following: where S-( (z, y) є R2 : 0 ST/2,0 < y ST/2). (a) Jls (cosz-s (b) fdl where y is the line segment from (2,-1,3) to (0, 1, 4) and f (x,y,z)-y+2 sin y) dA 3 marks 3 marks (c) Jc F dr where C is the unit circle centred at the origin, traversed once anticlockwise and F R2R2 is given by F(r,y)- (x2.x + y) 3 marks JJR eVEdA where R is the region enclosed by...
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