Jaдхdx dxQ3: Define the linear functional J: H()-R byتاریخ(v) = ½a(v, v) - (v)==Let u be the unique weak solution to a(u,v) = L(v) in H₁(2) and suppose thata(...) is a symmetric bilinear form on H() prove thata Buy v) =1- u is minimizer. 2- u is unique. 3- The minimizer J(u,) can be rewritten underJ(u)=u' Au-ub,algebraic formWhere A, b are repictively the stiffence matrix and the load vectorQ4: A) Answer only1-show that thelation to-Auf in N,u = 0 on a satisfies thestability Vulf and show that V(u-u,)||² = ||vu||2 - ||vu||2lu-ulls Chu||22- Prove thatWhere=1||ul|= a(u, u) = Vu. Vu dx + fu. udsB) Consider the bilinear forma(u, v) = (Au, Av) + (Vu, Vv) + (Vu, v) + (u, v)Show that a(u, v) continues and V- elliptic on H(2)(3)(0.0), (3.0)

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Answered: Ja дх dx dx Q3: Define the linear…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].
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(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of -Vf(x). Let 8 be the angle between Vf(x) and unit vector u. Then Duf = |Vf||---Select--- . Since the minimum value of ---Select--- is direction of u is ---Select--- ✓the direction of Vf (assuming Vf is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x¹y x²y³ decreases fastest at the point (1, -1). occurring, for 0 ≤ 0 < 2π, when 8 = the minimum value of Duf is -|Vf, occurring when the
Nedd help with d)

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