pleeee 10mi Q1: Let x(u, v) = (u2, uv, v2) be a simple surface. Prove that...
Let F = <z, 0, y> and let S be the oriented surface parametrized by G(u, v) = (u2 − v, u, v2) for 0 ≤ u ≤ 6, −1 ≤ v ≤ 4. Calculate the normal component of F to the surface at P = (24, 5, 1) = G(5, 1).
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be the hemisphere 2 F(x, y,z)-yitj+3z k. Calculate JJs F dS, the flux of F across S 7. Find the surface area of the surface r(u, u) = u ui + (u + u)j + (u-u) k, u2 +02-1 V/16-x2-y2 with upward orientation and let 8. Let S be...
[2 marks] Let V be the set of all ordered pairs of real numbers (u1, uv) with uj > 0. Consider the following addition and scalar multiplication operations on u = (u1, u2) and v = (v1, v2): u + v = (421, uz + v2), ku = (kuq, kuz) If the set V with the above operations satisfies Axiom 5 of a vector space (i.e., the existence of a negative element), what would be the negative of the vector...
Surface D is given by r(u, v) = u, v, u2 + v^2 , above the solid circle x 2 + y 2 ≤ 2, oriented upwards. The vector field F is (x^2 , xz, y2*z). Set up a double integral with the same value as Z ∂D F · dr, so that the integral is ready to evaluate. Do not evaluate either integral.
Exercise 3. Let u2= (5) C) V2 = V1 = and E u1, u2},F = {v1,v2} be two ordered bases for R2. Let also 5 (i) Find the coordinate vectors of [x]E and [x\f. (ii) Find the transition matrix S from the basis E to F. (ii) Verify that [x]f = S[r]E Exercise 3. Let u2= (5) C) V2 = V1 = and E u1, u2},F = {v1,v2} be two ordered bases for R2. Let also 5 (i) Find the...
dz Find when u = 0, v = 2, if z = sin (xy)+xsin (y), x=u2 +2V2, and y= uv. du az = du 1 = 0, V=2 (Simplify your answer.)
6. Let Ui, U,Un be independent Unif-2,0) random variables and Xa)in(U, U2, .., Un). Prove that X(a) converges in probability to -2
Exercise 4.5.3. Let G-(g g 1 be a group of order 2 and V a CG-module of Let u +202 +2,u2 2v1 - 2 +2vs,u vector space spanned by ui, for i-1,2,3 2v - 202 +vs, and hence U the (i) Prove that U is a CG-submodule of V fori 1,2,3, and that (ii) Let λ C and u-ul + U2 + λν3 V. Find the value(s) of λ for which the subspace U spanned by u is a CG-submodule...
79. Parametrised Surface 1. Consider the parametrised surface defined by: x 2u, y u2+ v (a) Find a vector normal to the surface in terms of u and v. (b) For what values of u and v is the surface smooth? (c) Find the equation of the tangent plane to the surface at (0,1, 1) 79. Parametrised Surface 1. Consider the parametrised surface defined by: x 2u, y u2+ v (a) Find a vector normal to the surface in terms...