a) Calculus f(x) = x, g(x) = e*, %3D (f, dx b) Let f(x) Example 5, 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in S.8) = f(x)g(x) dx, (f, to find the orthogonal projection of f onto g.

Q: 1) Find the work done by the vector field F(x, y, z) = (x², 4xy³, xy²) on a particle that %3D…

A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: Example Let F - xy i+ xy² j be a vector field in 2-space. Evaluate 4. xy? dx + xy? dy where C is the…

A: As per bartleby guidelines we only give first  questions answers.

Q: EXAMPLE 2 Show that the vector field F(x, y, z) = 4xzi + 5xyzj – y²k is not conservative. SOLUTION…

A:

Q: 3. (a) Show that f(r) = 1- are orthogonal elements of C[0, 1] with respect to the = z and g(a) inner…

A:

Q: Consider inner-product = :| f{x)g(x) dx in vector space C[- 17, n], the Fourier expansion of f(x) =…

A:

Q: Consider the vector space V = C([0, 1]) of complex-valued functions that are continuous on [0, 1].…

A: To establish the Cauchy- Schwarz inequality for the functions in C[0,1]= the space of complex valued…

Q: Consider the subspace W of D, given by W = span(sin x, cos x).                         (a) Show that…

A: Given Subspace W=span(sin x, cos x) of D{space of all differentiable functions} B={sin x, cos x}

Q: 3 , find a potential function for the vector field F by inspection or show that one does not exist.…

A:

Q: 7. Let C[-7,7] be the vector space of continuous function over [-n, 1] with an inner product 1 (5,9)…

A: C-π,π is a vector space of continuous function, with inner product <f,g>=1π∫-ππfxgxdx show…

Q: Consider the vector space V = span ({sin x, cos x}) over R with inner product (, ) : V x V → R…

A:

Q: a) Define a linear operator on R by performing the following operations in order: i) a shear in the…

A: As per policy, we are solving Question (a) only, Please repost the Question to get the solution for…

Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: : (2,5) in R². Let u = (1, –3) and v = (a) Find (u, v) and ||v|| with respect to the standard inner…

A: Inner product

Q: If V(F) is a vector space of polynomials in t of degree at most 3 and D be the differentiation…

A: Given D is differentiation transformation. The basis is 1, t, t2, 1+t3. T(1)=ddt1=0…

Q: Let f(x) = sin x and  g(x) = sin x + cos x  in the vector space C, [0, 2π] with the inner product…

A: Given that, f(x) = sin x and  g(x) = sin x + cos x Consider an inner product in them for f , g C ∈…

Q: Consider the vector space C[-a, 7] with inner product / f(r)g(x)dr. (a) Show that the vectors f(x) =…

A:

Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

A:

Q: 3. Let F be the vector field (ze²z + yz², xz², xe*z + 2xyz)

A:

Q: Consider inner-product = ] f(x)g(x) dx defined for vector space C[-1, 1] , then for the -1 function…

A:

Q: Find the Wronskian Determinate of g(x) = sinx f(x) = x and %3D %3D اخترأحد الخيارات W(f, g) = x cosx…

A:

Q: Find vectors x and y in R2 that are orthonormal with respect to the inner product (u, v) = 3u₁v₁ +…

A: The given problem is related with inner product. We have to find the vectors x and y in ℝ2 that are…

Q: 5. Find the standard matrix corresponding to the linear map T : R2 - R? given by T(r, y) = (r+y,…

A:

Q: Let v=(-2,3,0,6), find all scalars k such that |kv|=5 LUTION

A:

Q: Consider inner-product =| f(x) g(x) dx in vector space C[- n, 7], the Fourier expansion of f(x) = x…

A:

Q: 3 Let A be the following linear functional on C([0, 1]): xf (Va) da, f e C([0, 1]). Prove that A is…

A:

Q: Consider inner-product = | f(x)g(x) dx defined for vector space C[-1, 1] , -1 then for the function…

A:

Q: 3. Given the functions f, (x) = x and f2(x) = sin2x on the interval - %3D (a) Find the norm of f, on…

A:

Q: Let f: C → C be differentiable at zo. Considering ƒ as a map R? → R², prove that f'(zo) = kR where k…

A: (a)

Q: Prove that F = 2(x+y sin z)î+2x sin zj+ 2xy cos zk is a conservative vecto field by finding a scalar…

A: The objective is to find Given that, Since, so,

Q: Let F(x, y) = (4 cos(y), 2 sin(y)). Compute the flux $ F - n ds of F across the boundary of the…

A: By using Green Theorem, ∮F.n ds…

Q: Show that F has a fixed point (b) La B væte 3 x 3-matrisen og la F = By+0 Vis at F har et fikspunkt.…

A:

Q: (b) Verify Green's Theorem for a vector field F(x,y) = xy²i – yx²j on a disc with center (0,0) and…

A:

Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2, v3).…

A:

Q: Let B = {1, x, x², x³, xª}, which is a basis for the vector space of polyno- mials W = P4(R). Let V…

A: Let the best approximation of the function f(x)=sinx+cosx is g(x), where  g(x)=∑i=15f(x),eiei…

Q: 1. (a) Define Linear transformation of vector spaces. Let T : R? →R be denoted by T(x, y)= (x cos 0…

A: T is a linear transformation, if the following two properties hold: T(x+y) = T(x)+T(y)T(cx) = cT(x)…

Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

A:

Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1] , then for the function…

A:

Q: In C[-1,1], with the standard inner product f(x)=ex and f(x)=e−x are orthogonal. true or false?

A: true please see explanation in step 2

Q: Consider inner-product = | f(x) g(x) dx defined for vector space C[-1, 1] , then for the function…

A: The solution is given as

Q: Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (f, g) = f(x)g(x) dx.…

A:

Q: Consider inner-product = | f(x)g(x) dx in vector space C[- n, T], the Fourier expansion of f(x) = x…

A:

Q: Use the Fourier transform to solve the IBVP U = a² (urz + Uyy), 0 < x < 0, 0 < y < o. u (x, y,0) = f…

A:

Q: Curl f a vedor point field = xi+ itzk is © 2x it 2yj+2zk 2x+ 24+ 2

A: Curl of a vector field F→=f1i^+f2j^+f3k^ is given by: Curl (F)=i^j^k^∂∂x∂∂y∂∂zf1f2f3 Here .…

Q: Let f: C → C be differentiable at zo. Considering f as a map R? → R², that f'(zo) = kR where k is a…

A: (a)

Q: Find the orthogonal projection of f onto g. Use the inner product in C[a, b] (, 9) = [". f(x)g(x)…

A:

Q: Find the orthogonal projection of f onto g. Use the inner product in Ca, b] (f, 9) = ["«x»9(x) dx.…

A:

Q: Let f: C → C be differentiable at zo. Considering f as a map R? –→ R², prove that f'(zo) = kR where…

A: (a)

Q: vVr ITe uruit 10 dEscribe EUCN TraNSTOPmUTON. 本y A 9) A' Z' Z'. K N' K' A n'. K. K' Rule: 12)

A: The given figure AZNK translated into A'Z'N'K'

Q: Let R2[x]be the vector space of real polynomials of degree < 2. Let f: R2x] - R2x]be the…

A: