(a) By the Heine-Borel Theorem, show that R2 is not compact and
the
sphere
S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1}
is compact in R3.
(b) Show that R2 and S2 is not homeomorphic. (i.e. no continuous
bi-
jective function f between R2 and S2 such that the inverse function
f−1 is continuous).
Homework Answers
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(a) By the Heine-Borel Theorem, show that R2 is not compact and the sphere S2 ={(x,y,z)∈R3 :x2 +y2 +z2 =1} is compact in...
Q.7, as question above 7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz...
Q.7, as question above
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
A2) Let Sl be the unit circle z2 + y2-l in R2. Let S2 be the unit sphere z2 + y2 + z2-l in R. Let...
A2) Let Sl be the unit circle z2 + y2-l in R2. Let S2 be the unit sphere z2 + y2 + z2-l in R. Let Sn be the unit hypersphere x| + z +--+ z2+1-1 in Rn+1 (a) Write an iterated double integral in rectangular coordinates that expresses the area inside S1. Write an iterated triple integral in rectangular coordinates that expresses the volume inside S2. Write an iterated quadruple integral in rectangular coordinates that expresses the hypervolume inside...
2. Consider the conical surface S={(x,y,z)∈R3 : x2 + y2 = z2, 0 ≤ z ≤...
2. Consider the conical surface
S={(x,y,z)∈R3 : x2 + y2 =
z2, 0 ≤ z ≤ 1},
and the vector field
(a) Carefully sketch S, and identify its boundary ∂S.
(b) By parametrising S appropriately, directly compute the flux
integral
S (∇ × f) · dS.
(c) By computing whatever other integral is necessary (and
please be careful about explaining any orien- tation/direction
choices you make), verify Stokes’ theorem for this case.
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 +...
The gravitational field F(x,y,z) =cx /(x2 + y2 + z2)3/2 e1+ cy /(x2 + y2 + z2)3/2 e2+ cz/ (x2 + y2 + z2)3/2 e3 is a gradient field, where c is a constant, such that the field is rotation free. If we define f(x,y,z) = −c /(x2 + y2 + z2)1/2 , then show that (a) F = grad(f). (b) curl(F) = 0.
You have been asked to find the points on the sphere x2 + y2 + z2...
You have been asked to find the points on the sphere x2 + y2 + z2 = 36 that are closest to and farthest from the point (1, 2, 2). Then which of the following is incorrect from the following: Select one: A. The point on the sphere farthest to the point (1,2,2) is (-2,-4,-4) B. The point on the sphere closest to the point (1,2,2) is (2, 4,4) C. The solutions to the question can be found by solving...
5. Evaluate JSF dS, where and S is the top half of the sphere x2 + y2 + z2-1. Note that S is not a closed surface. Ther...
5. Evaluate JSF dS, where and S is the top half of the sphere x2 + y2 + z2-1. Note that S is not a closed surface. Therefore you must first find a surface Sı such that you can (a) Evaluate the flux of F across S (b) Use the divergence theorem on SUSi
5. Evaluate JSF dS, where and S is the top half of the sphere x2 + y2 + z2-1. Note that S is not a closed...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
1 point) Use Stoke's Theorem to evaluate (▽ × F)·dS where M is the hemisphere x2 + y2 + z2-16, x ...
1 point) Use Stoke's Theorem to evaluate (▽ × F)·dS where M is the hemisphere x2 + y2 + z2-16, x > 0, with the normal in the direction of the positive x direction, and F (x6,0,yl) Begin by writing down the "standard" parametrization of ЭМ as a function of the angle θ (denoted by "t" in your answer) a F-dsf(0) de, where f(θ) = The value of the integral is (use "" for theta).
1 point) Use Stoke's Theorem...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a)...
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
Q.7, as question above
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
7. Verify the divergence theorem for F(x, y,z) - by the sphere x2 + y2 + Z2-4. 4xöx +yy +4zőz and V is the region bounded (15 points)
A2) Let Sl be the unit circle z2 + y2-l in R2. Let S2 be the unit sphere z2 + y2 + z2-l in R. Let Sn be the unit hypersphere x| + z +--+ z2+1-1 in Rn+1 (a) Write an iterated double integral in rectangular coordinates that expresses the area inside S1. Write an iterated triple integral in rectangular coordinates that expresses the volume inside S2. Write an iterated quadruple integral in rectangular coordinates that expresses the hypervolume inside...
2. Consider the conical surface
S={(x,y,z)∈R3 : x2 + y2 =
z2, 0 ≤ z ≤ 1},
and the vector field
(a) Carefully sketch S, and identify its boundary ∂S.
(b) By parametrising S appropriately, directly compute the flux
integral
S (∇ × f) · dS.
(c) By computing whatever other integral is necessary (and
please be careful about explaining any orien- tation/direction
choices you make), verify Stokes’ theorem for this case.
You have been asked to find the points on the sphere x2 + y2 + z2 = 36 that are closest to and farthest from the point (1, 2, 2). Then which of the following is incorrect from the following: Select one: A. The point on the sphere farthest to the point (1,2,2) is (-2,-4,-4) B. The point on the sphere closest to the point (1,2,2) is (2, 4,4) C. The solutions to the question can be found by solving...
5. Evaluate JSF dS, where and S is the top half of the sphere x2 + y2 + z2-1. Note that S is not a closed surface. Therefore you must first find a surface Sı such that you can (a) Evaluate the flux of F across S (b) Use the divergence theorem on SUSi
5. Evaluate JSF dS, where and S is the top half of the sphere x2 + y2 + z2-1. Note that S is not a closed...
r 37. Singular radial field Consider the radial field (x, y, z) F (x2 + y2 + z2)1/2" a. Evaluate a surface integral to show that SsFonds = 4ta?, where S is the surface of a sphere of radius a centered at the origin. b. Note that the first partial derivatives of the components of F are undefined at the origin, so the Divergence Theorem does not apply directly. Nevertheless, the flux across the sphere as computed in part (a)...
1 point) Use Stoke's Theorem to evaluate (▽ × F)·dS where M is the hemisphere x2 + y2 + z2-16, x > 0, with the normal in the direction of the positive x direction, and F (x6,0,yl) Begin by writing down the "standard" parametrization of ЭМ as a function of the angle θ (denoted by "t" in your answer) a F-dsf(0) de, where f(θ) = The value of the integral is (use "" for theta).
1 point) Use Stoke's Theorem...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Problem 1: Consider the following problem x+y+1=1 x2 +y2+z2 =1 max f(x ,y,z)=er+y+1 subject to (a) Solve the problem. (b) Replace the constraints byx+y+1=1.02 and x2+y2+Z2-0.98. What is the approximate change in the optimal value of the objective function? (c) Classify the candidate points for optimality in the local optimization problem.
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