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Question 4. Take the curve y cosh r in the r-y plane, and revolve it around the z axis. The resul...
(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the a...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
Let S be the ‘football’ surface formed by rotating the curve y = 0, x = cos z for z ∈ [−π/2, π/2], around the z-axis. Find a parametrization for S, and compute its surface area. Please answer in fu...
Let S be the ‘football’ surface formed by rotating the curve y =
0, x = cos z for z ∈ [−π/2, π/2], around the z-axis. Find a
parametrization for S, and compute its surface area. Please answer
in full With full instructions.
Let S be the 'football, surface formed by rotating the curve y = 0, x-cosz for-E-π/2, π/2], around the z-axis. Find a parametrization for S, and compute its surface area 3
Let S be the 'football, surface...
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 →...
Question 10 (11 marks) In your sketches in this question, label any axis intercepts with their values. Let S be the sur...
Question 10 (11 marks) In your sketches in this question, label any axis intercepts with their values. Let S be the surface in R3 given by z 2+ +yfor (r, y) R2 (a) Find an expression for the level curve of this surface when zc. For what value(s) of c does the level curve exist? (b) Sketch the level curve for c 2, and the level curve for c 5 (c) Sketch the cross section of the surface in the...
Instructions: Show all your work for FULL credit. Calculators are NOL final answer. Neatness is highly appreciated. 1. A region R, bounded by y 2x, y 6-x, and x-axis, is rotated around the y-axis. Sk...
Instructions: Show all your work for FULL credit. Calculators are NOL final answer. Neatness is highly appreciated. 1. A region R, bounded by y 2x, y 6-x, and x-axis, is rotated around the y-axis. Sketch the region R, in the box a) 15 strip/slice you will use to find the volume of the solid of revolution. b) Write the definite integral that gives a X the volume of the solid of revolution. (DO NOT evaluate the integral.) Find the circumference...
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...
Use Stokes' theorem to find the work done by the force field F(z, y, z)-<-r, z, y > along the positively oriented curve of intersection of the cylinder 2+y 1 and the plane 3x +z 4 9....
Use Stokes' theorem to find the work done by the force field F(z, y, z)-<-r, z, y > along the positively oriented curve of intersection of the cylinder 2+y 1 and the plane 3x +z 4 9.
Use Stokes' theorem to find the work done by the force field F(z, y, z)- along the positively oriented curve of intersection of the cylinder 2+y 1 and the plane 3x +z 4 9.
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth...
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
5 pts) Consider the region bounded by the curves y 9, y and r 1 r-+64 If this region is revolved ...
5 pts) Consider the region bounded by the curves y 9, y and r 1 r-+64 If this region is revolved around the x - axis, the volume of the resulting solid can be computed in (at least) two different ways using integrals. (Sketching the graph of the situation m (a) First of all it can be computed as a single integral h(r)dr where o and This method is commonly called the method of Enter 'DW' for Disks/Washers or 'CS...
Please, include the explanations to your solution! Begin with the paraboloid z #x2 + y2, for 0 s z s 64, and slice it with the plane y-0. Let S be the surface that remains for y 20 (including the...
Please, include the explanations to your solution!
Begin with the paraboloid z #x2 + y2, for 0 s z s 64, and slice it with the plane y-0. Let S be the surface that remains for y 20 (including the planar surface in the xz-plane) (see figure). Let C be the semicircle and line segment that bound the cap of S in the plane z 64 with counterclockwise orientation. Let F (4z 3y,4x+3z,4y+3x. Complete parts (a) through (c) below 64...
(4) (a) Determine the standard matrix A for the rotation r of R
3 around the z-axis through the angle π/3 counterclockwise. Hint:
Use the matrix for the rotation around the origin in R 2 for the
xy-plane. (b) Consider the rotation s of R 3 around the line
spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a
basis of R 3 for which the matrix [s]B,B is equal to A from (a).
(c) Give...
Let S be the ‘football’ surface formed by rotating the curve y =
0, x = cos z for z ∈ [−π/2, π/2], around the z-axis. Find a
parametrization for S, and compute its surface area. Please answer
in full With full instructions.
Let S be the 'football, surface formed by rotating the curve y = 0, x-cosz for-E-π/2, π/2], around the z-axis. Find a parametrization for S, and compute its surface area 3
Let S be the 'football, surface...
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 →...
Question 10 (11 marks) In your sketches in this question, label any axis intercepts with their values. Let S be the surface in R3 given by z 2+ +yfor (r, y) R2 (a) Find an expression for the level curve of this surface when zc. For what value(s) of c does the level curve exist? (b) Sketch the level curve for c 2, and the level curve for c 5 (c) Sketch the cross section of the surface in the...
Instructions: Show all your work for FULL credit. Calculators are NOL final answer. Neatness is highly appreciated. 1. A region R, bounded by y 2x, y 6-x, and x-axis, is rotated around the y-axis. Sketch the region R, in the box a) 15 strip/slice you will use to find the volume of the solid of revolution. b) Write the definite integral that gives a X the volume of the solid of revolution. (DO NOT evaluate the integral.) Find the circumference...
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...
Use Stokes' theorem to find the work done by the force field F(z, y, z)-<-r, z, y > along the positively oriented curve of intersection of the cylinder 2+y 1 and the plane 3x +z 4 9.
Use Stokes' theorem to find the work done by the force field F(z, y, z)- along the positively oriented curve of intersection of the cylinder 2+y 1 and the plane 3x +z 4 9.
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
5 pts) Consider the region bounded by the curves y 9, y and r 1 r-+64 If this region is revolved around the x - axis, the volume of the resulting solid can be computed in (at least) two different ways using integrals. (Sketching the graph of the situation m (a) First of all it can be computed as a single integral h(r)dr where o and This method is commonly called the method of Enter 'DW' for Disks/Washers or 'CS...
Please, include the explanations to your solution!
Begin with the paraboloid z #x2 + y2, for 0 s z s 64, and slice it with the plane y-0. Let S be the surface that remains for y 20 (including the planar surface in the xz-plane) (see figure). Let C be the semicircle and line segment that bound the cap of S in the plane z 64 with counterclockwise orientation. Let F (4z 3y,4x+3z,4y+3x. Complete parts (a) through (c) below 64...
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Question 501 pts
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