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3.2. Show that als) (1 )2 (1- s)3/2, is a unit speed curve and compute its...
x: U- R3 and S its intrinsic normal. 3. Let y be a unit speed curve...
x: U- R3 and S its intrinsic normal. 3. Let y be a unit speed curve in a coordinate patch ".ξροη, 2 2 Lifh and S Derive the equations i, j- 1 k1 i, j= 1
x: U- R3 and S its intrinsic normal. 3. Let y be a unit speed curve in a coordinate patch ".ξροη, 2 2 Lifh and S Derive the equations i, j- 1 k1 i, j= 1
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin...
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
Prove that a unit speed curve with k and tour constant is a helix circular. unit...
Prove that a unit speed curve with k and tour constant is a
helix circular.
unit speed 3(10pts). Prove that with is and constant curve a T circular helix.
unit speed 3(10pts). Prove that with is and constant curve a T circular helix.
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature...
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
1 ,for 1 x 2 4x 2. Compute the length of the curve f(x) 3 1...
1 ,for 1 x 2 4x 2. Compute the length of the curve f(x) 3
1 ,for 1 x 2 4x 2. Compute the length of the curve f(x) 3
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...
4.25. Combine the previous result with Proposition 4.10 to prove that if a(s) is a unit speed cur...
4.25. Combine the previous result with Proposition 4.10 to prove that if a(s) is a unit speed curve with K,-0, τ 0, then α(s) lies on a sphere if and only if τ/K- . (K7sK')' (or τρ-:-(p'/t)'). PROPOSITION 4.10. Let α(s) be a unit speed curve 0. If τ whose lies sphere on a image 0, then of radius r and center m. Then K where ρ :-1/K and σ-1/τ. Hence rz-
4.25. Combine the previous result with Proposition 4.10...
problem 3 pls Problem 3. Consider the curve {> 1, y = 1/x}. Compute the length...
problem 3 pls
Problem 3. Consider the curve {> 1, y = 1/x}. Compute the length of the part of this curve lying to the left of the line x = a for any a > 1. Show that the length of the whole curve is thus infinite. Compute the area of the surface obtained by rotating this curve about about the x axis by computing the corresponding improper integral; it should be infinite. What is the area of the...
9. Given Firm 1's best-response curve, show how Firm 2's best-response curve shifts on a graph...
9. Given Firm 1's best-response curve, show how Firm 2's best-response curve shifts on a graph if its marginal cost changes from m to m+x. Show how the Nash-Cournot equilibrium changes as Firm 2's marginal cost increases. O O Initially, assume the marginal cost for both firms is m, resulting in the best-response functions shown in the graph. 3 Now assume that the second firm's marginal cost of production changes from m to m + x 1.) Using the line...
x: U- R3 and S its intrinsic normal. 3. Let y be a unit speed curve in a coordinate patch ".ξροη, 2 2 Lifh and S Derive the equations i, j- 1 k1 i, j= 1
x: U- R3 and S its intrinsic normal. 3. Let y be a unit speed curve in a coordinate patch ".ξροη, 2 2 Lifh and S Derive the equations i, j- 1 k1 i, j= 1
Question 1. Let y : R -> R' be the parametrised curve 8 (t)= 1+ sin t Cost 5 Cos (a) (2 marks) Show that y is unit speed (7 marks) Find, at each point on the curve, the principal tangent T, principal normal (b) N, binormal B, curvature K, and torsion 7. (c) (3 marks) Show directly that T, N, B satisfy the Frenet-Serret frame equations (d) (3 marks) Show that the image of y lies in a plane...
7. Let a be a unit-speed curve in M CR?. Instead of the Frenet frame field on a, consider the Darboux frame field T, V, U—where T is the unit tangent of a, U is the surface normal restricted to a, and V = U * T (Fig. 5.34). (a) Show that T' = gV + kU V' =-gT + tU, U' = -KT - tv, 263/518 where k = S(T) · T is the normal curvature k(T) of M...
Prove that a unit speed curve with k and tour constant is a
helix circular.
unit speed 3(10pts). Prove that with is and constant curve a T circular helix.
unit speed 3(10pts). Prove that with is and constant curve a T circular helix.
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
1 ,for 1 x 2 4x 2. Compute the length of the curve f(x) 3
1 ,for 1 x 2 4x 2. Compute the length of the curve f(x) 3
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...
4.25. Combine the previous result with Proposition 4.10 to prove that if a(s) is a unit speed curve with K,-0, τ 0, then α(s) lies on a sphere if and only if τ/K- . (K7sK')' (or τρ-:-(p'/t)'). PROPOSITION 4.10. Let α(s) be a unit speed curve 0. If τ whose lies sphere on a image 0, then of radius r and center m. Then K where ρ :-1/K and σ-1/τ. Hence rz-
4.25. Combine the previous result with Proposition 4.10...
problem 3 pls
Problem 3. Consider the curve {> 1, y = 1/x}. Compute the length of the part of this curve lying to the left of the line x = a for any a > 1. Show that the length of the whole curve is thus infinite. Compute the area of the surface obtained by rotating this curve about about the x axis by computing the corresponding improper integral; it should be infinite. What is the area of the...
9. Given Firm 1's best-response curve, show how Firm 2's best-response curve shifts on a graph if its marginal cost changes from m to m+x. Show how the Nash-Cournot equilibrium changes as Firm 2's marginal cost increases. O O Initially, assume the marginal cost for both firms is m, resulting in the best-response functions shown in the graph. 3 Now assume that the second firm's marginal cost of production changes from m to m + x 1.) Using the line...
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