Q3: Define the linear functional J: H(2) R by 1(v) = a(v. v) - L(v) Let u be the unique weak solution to a(u,v) = L(v) in H() and suppose that a(...) is a symmetric bilinear form on H(2) prove that 1- u is minimizer. 2- u is unique. 3- The minimizer J(u,) can be rewritten under algebraic form u Au-ub. J(u)=u'Au- Where A. b are repictively the stiffence matrix and the load vector

Consider the points (xo, yo) (x₁, y₁) (x2, Y2) (X3, Y3) (X4, Y4) + = X = = = = (-3,-4), (-1, 1), (0, 1), Find the best linear fit for these points, i.e., the function of the form p(x) = a + bx such that the sum of the squared errors \p(x;) — y¡ |² is as small as possible. Answer: p(x) = (2,5), (3,9).

CAN YOU EXPLAIN THE DIFFERENCE BETWEEN i and ii AND CAREFULLY EXPLAIN HOW YOU GO ABOUT CONSTRUCTING AN EQUATION FOR EACH, THANKS

Use the Wronskian to determine if the given functions are linearly independent on the indicated interval.   ​f(x)=13​, ​g(x)=3x​, ​h(x)=x^2​; the real line   Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.   ​(Simplify your​ answer.)   A.The Wronskian​ W(f, g,​h)=___?. As W is identically 0 on the real line​ f(x), g(x) and​ h(x) are linearly independent.   B. The Wronskian​ W(f, g,​h)=___? As W is never 0 on the real line​ f(x), g(x) and​ h(x) are linearly independent.   C.The Wronskian​ W(f, g,​h)=___? As W is identically 0 on the real line​ f(x), g(x) and​ h(x) are linearly dependent.   D. The Wronskian​ W(f, g,​h)=___?. As W is never 0 on the real line​ f(x), g(x) and​ h(x) are linearly dependent.

Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.

Write the Linear Approximation to f(x, y) = x(1 + y)-¹ at (a, b) = (8, 1) in the form f(a+h, b+k) f(a, b) + fx(a, b)h + fy(a, b)k (Give an exact answer. Use decimal notation and fractions where needed.) f(8 + h, 1+k) ≈ Use it to estimate 7.97 2.05 (Use decimal notation. Give your answer to three decimal places.) 7.97 2.05 22 Compare this approximation with the value obtained using the calculator. Calculate the percentage error obtained by using the approximation. Percentage error is the absolute value of the ratio of the error to the actual value of the expression, multiplied by 100%. (Use decimal notation. Give your answer to three decimal places.) the percentage error: %

Let T (V₁, V2, V3) w = (4, 1,-1). = (2v1 + V2, 2v2 - 3v₁, V₁-V3). Find T(-4,5, 1) and pre-image of

Let F = (2xz + y², 2xy, x² + 1), if Vf= F, find f(1, 1, 1) - f(0,0,0)

What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) do not hold?

Q3: Define the linear functional J: H₁(2) R by ¡(v) = a(v, v) - L(v) Л Let u be the unique weak solution to a(u,v) = L(v) in H(2) and suppose that a(...) is a symmetric bilinear form on H(2) prove that 1- u is minimizer. 2- u is unique. 3- The minimizer J(u) can be rewritten under 1(u) = u Au-ub, algebraic form 1 2 Where A, b are repictively the stiffence matrix and the load vector Q4: A) Answer 1- show that the solution to -Au = f in A, u = 0 on a satisfies the stability Vullfll and show that ||V(u u)||||||2 - ||vu||2 2- Prove that Where lu-ul Chuz - !ull = a(u, u) = Vu. Vu dx + fu. uds B) Consider the bilinea forta Л a(u, v) = (Au, Av) (Vu, Vv + (Vu, v) + (u,v) Show that a(u, v) continues and V- elliptic on H(2)