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Let M be the unit sphere, x the spherical coordinate, and y the inverse stereographic projection...
2. Recall the usual stereographic projection of C to the Riemann sphere C, where a point...
2. Recall the usual stereographic projection of C to the Riemann sphere C, where a point z in the plane corresponds to a point Z on the sphere when the line (in R3) joining the north pole N to2z intersects the sphere at Z. Now consider the (inverse) stereo- graphic projection taking a point Z on the sphere back to some w in the plane by reversing the process, but instead using the line oining Z with the south pole...
Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the South Pôle in Antartica at the origin. If D C R2 is mapped to A S2, the...
Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the South Pôle in Antartica at the origin. If D C R2 is mapped to A S2, then lAf dA-JD? References William Briggs, Lyle Cochran, and Bernard Gillett. Calculus. Pearson, Boston, MA, second edition, 2016. With the assistance of Eric Schulz
Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the...
Please write neat and explain thank you. This problem concerns embedding the complex plane C with...
Please write neat and explain thank you.
This problem concerns embedding the complex plane C with elements zx iy in the Riemann sphere defined in 3-dimensional space R' with coordinates (X,Y,Z) as the set of points satisfying X2 + Y2+22 = 1, which is known as the unit sphere and denoted by S2,or in the context of stereographic projection of the complex plane into the sphere, often referred to as the extended complex plane and denoted by C. We identify...
t5.14. Let x and y be two different coordinate patches for part of a surface M Let X Xx, X, and Y = Y'x Y'...
t5.14. Let x and y be two different coordinate patches for part of a surface M Let X Xx, X, and Y = Y'x Y'y. be two vector fields. Define symbols Z* and Z by χι ΣΓ/Υιχ | and Prove that 2 EZ*(dv|8u*). (Hint: Problem 4.11.) This proves thatZx £ Zy, defines a vector field Z = VxY, called the covariant derivative of Y with respect to X. This is one of the most fundamental concepts of modern differential geometry....
Let us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR a...
We were unable to transcribe this imageLet us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points)
Let us denote...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on an inclined plane with inclination angle ? in the gravitational field g. Follow the steps below to find the velocity V of the center of mass of the sphere as a function of time if the sphere is initially at rest. Bold font represents vectors. There exists a reaction force R at the point of contact between the sphere and the plane. The equations...
Find the x- and y-components of B⃗ in the coordinate system shown in (Figure 1) (b). Let...
Find the x- and y-components of
B⃗ in the coordinate system shown in (Figure 1)
(b).
Let B = (9.0 m, 50 degree counterclockwise from the vertical upward direction). Find the x- and y-components of B in the coordinate system shown in (Figure 1) (a). Enter your answers numerically separated by a comma. B_x, B_y = - 6.9, 5.8 m Here we learn how to find components of a vector given its magnitude and direction. Find the x- and y-components of...
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and ...
please help with Q1 and 3
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and below the cone z -Vx2 + y2 (a) Sketch the region V (b) Calculate the volume of V by using spherical coordinates. (c) Find the surface area of the part of V that lies on the sphere z2 y 24, by calculatinga surface integral. (d) Verify your solution to (c) by calculating the surface integral...
Q1. Sketch and find a parameterization of the following surfaces (a) (Spherical cap) The portion of the sphere by *+y+-16 cut by the vertical plane y 3, containing the point (0,4,0) (b) (Circular cyl...
Q1. Sketch and find a parameterization of the following surfaces (a) (Spherical cap) The portion of the sphere by *+y+-16 cut by the vertical plane y 3, containing the point (0,4,0) (b) (Circular cylinder band) The portion of the cylinder y+z+5)2 25 between the planes x--2 and x-2.
Q1. Sketch and find a parameterization of the following surfaces (a) (Spherical cap) The portion of the sphere by *+y+-16 cut by the vertical plane y 3, containing the point (0,4,0) (b)...
GNOMONIC PROJECTION x(m)x (on scale) 90 75 60 45 30 35 0 1707104 3678299 6371000 1034896 9098731 ...
GNOMONIC PROJECTION x(m)x (on scale) 90 75 60 45 30 35 0 1707104 3678299 6371000 1034896 9098731 0.000 0.016 0.033 0.058 0.100 0.083 (meters) 6371000 9.09091E-09 Scale 1 110000000 MERCATOR PRJECTION p Y (mY (on scale) -7.07323E-101 15 30 45 60 75 1687309.96 499629.446 5615231.123 8390338.761 12917772.21 0.000 0.014 0.029 0.047 0.070 0.108 1.982 90 237838646.6 35 0.035 (meters) 4159221.849 6371000 8.33333E-09 Scale 1 120,000,000 2"TT R (equator length) 0030173.590.33358478 (meters) on earth on scale Draw on a letter size...
2. Recall the usual stereographic projection of C to the Riemann sphere C, where a point z in the plane corresponds to a point Z on the sphere when the line (in R3) joining the north pole N to2z intersects the sphere at Z. Now consider the (inverse) stereo- graphic projection taking a point Z on the sphere back to some w in the plane by reversing the process, but instead using the line oining Z with the south pole...
Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the South Pôle in Antartica at the origin. If D C R2 is mapped to A S2, then lAf dA-JD? References William Briggs, Lyle Cochran, and Bernard Gillett. Calculus. Pearson, Boston, MA, second edition, 2016. With the assistance of Eric Schulz
Problem 3 Parametrize the unit sphere S2 with a stereographic projection: a stereographic map of a spherical Earth with the...
Please write neat and explain thank you.
This problem concerns embedding the complex plane C with elements zx iy in the Riemann sphere defined in 3-dimensional space R' with coordinates (X,Y,Z) as the set of points satisfying X2 + Y2+22 = 1, which is known as the unit sphere and denoted by S2,or in the context of stereographic projection of the complex plane into the sphere, often referred to as the extended complex plane and denoted by C. We identify...
t5.14. Let x and y be two different coordinate patches for part of a surface M Let X Xx, X, and Y = Y'x Y'y. be two vector fields. Define symbols Z* and Z by χι ΣΓ/Υιχ | and Prove that 2 EZ*(dv|8u*). (Hint: Problem 4.11.) This proves thatZx £ Zy, defines a vector field Z = VxY, called the covariant derivative of Y with respect to X. This is one of the most fundamental concepts of modern differential geometry....
We were unable to transcribe this imageLet us denote the volume and the surface area of an n-dimensional sphere of adius R as V(OR)-VR and S(R)-S.),respectively (a) Find the relation between V(0) and S 1) (b) Calculate the Gaussian integral 3. (c) Calculate the same integral in spherical coordinates in terms of the gamma function re)-e'd (d) Obtain the closed forms of S,,(1) and V(1) (e) Calculate r5) and S.,0), p.(1) for n-1, 2, 3. (40 points)
Let us denote...
Problem 2 (10 pt.) A homogeneous sphere of mass m and radius b is rolling on an inclined plane with inclination angle ? in the gravitational field g. Follow the steps below to find the velocity V of the center of mass of the sphere as a function of time if the sphere is initially at rest. Bold font represents vectors. There exists a reaction force R at the point of contact between the sphere and the plane. The equations...
Find the x- and y-components of
B⃗ in the coordinate system shown in (Figure 1)
(b).
Let B = (9.0 m, 50 degree counterclockwise from the vertical upward direction). Find the x- and y-components of B in the coordinate system shown in (Figure 1) (a). Enter your answers numerically separated by a comma. B_x, B_y = - 6.9, 5.8 m Here we learn how to find components of a vector given its magnitude and direction. Find the x- and y-components of...
please help with Q1 and 3
1. Let V be the solid region in R3 that lies within the sphere 2+y+z2-4, above the zy-plane, and below the cone z -Vx2 + y2 (a) Sketch the region V (b) Calculate the volume of V by using spherical coordinates. (c) Find the surface area of the part of V that lies on the sphere z2 y 24, by calculatinga surface integral. (d) Verify your solution to (c) by calculating the surface integral...
Q1. Sketch and find a parameterization of the following surfaces (a) (Spherical cap) The portion of the sphere by *+y+-16 cut by the vertical plane y 3, containing the point (0,4,0) (b) (Circular cylinder band) The portion of the cylinder y+z+5)2 25 between the planes x--2 and x-2.
Q1. Sketch and find a parameterization of the following surfaces (a) (Spherical cap) The portion of the sphere by *+y+-16 cut by the vertical plane y 3, containing the point (0,4,0) (b)...
GNOMONIC PROJECTION x(m)x (on scale) 90 75 60 45 30 35 0 1707104 3678299 6371000 1034896 9098731 0.000 0.016 0.033 0.058 0.100 0.083 (meters) 6371000 9.09091E-09 Scale 1 110000000 MERCATOR PRJECTION p Y (mY (on scale) -7.07323E-101 15 30 45 60 75 1687309.96 499629.446 5615231.123 8390338.761 12917772.21 0.000 0.014 0.029 0.047 0.070 0.108 1.982 90 237838646.6 35 0.035 (meters) 4159221.849 6371000 8.33333E-09 Scale 1 120,000,000 2"TT R (equator length) 0030173.590.33358478 (meters) on earth on scale Draw on a letter size...
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