7.Let M be a subspace of linear space X over field F. Define M° ={f eX':f(x)=0 \xEM} b. dim M+dim MⓇ=dim X Prove or disprove a. M° is a subspace of X' C. M = M

Q: 4. Let X = x Ә ду the Lie bracket Ә у and Y дх [X, Y]. Ә = z= +y дх Ә be differentiable vector…

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Q: 1 ExL =5 (ex-5) f(x). g(»)dx (F-9) ニ Let v be the vector space of oll continous real valued function…

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Q: 27. Let S be the set of all vectors both of y in R' such that x + y -z = 0 and 2x – 3y + 2z = 0.…

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Q: Consider F(x, y) = (y² − 2xy) i + (2xy — x²) j. (a) Show that F is a conservative vector field in…

A: the given vector field is F(x,y)=(y2-2xy)i+(2xy-x2)j we have to show that the vector field is…

Q: Suppose that Tis the projection on W along some subspace W'. Prove that W is T -invariant and that T…

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Q: Evaluate the work done between point 1 and point 2 for the conservative field F. F 3D6xi+ 6уј+ 6z k;…

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Q: 5. Let F(x, y) = square region with vertices (0,0), (1,0), (1, 1), and (0, 1) oriented…

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Q: 6-4 - Let X and Y be normed spaces and Z be a closed subspace of X. (a) Let F E BL(X,Y) and Z C…

A: See that this is a post with multiple questions, i.e., two or more unrelated questions. Here,…

Q: suppose Vn is the vector space of polynomials of degree <n-1 and with real cofficients. Determine…

A: As per the question we are given a vector space Vn of polynomials with degree ≤ n-1 and with real…

Q: Suppose F = (P(x, y, z), Q(x, y, z,), R(x, y, z)), is conservative vector field. Here P, Q, R are…

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Q: Find the divergence of the vector field v = (x²y² - z3)i + 2xyz j + e*yzk. a) 2xy2 + 2xz + xy e*yz…

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Q: Find the divergence of the vector field F = div F =

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Q: 2. Consider the vector-valued function R(t) = Show that is continuous at t = 1. (e¹-², 1-1², (2-1)),…

A: We use the sequential definition of continuity to show that $R(t)$ is continuous at t=1. The…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: 11. LetF(x, y) = (ye™ – 6x))i + (xe™v + 4)7 (a) Show that F~ is a conservative vector field. (b)…

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Q: 2. F(t)= 41³ +21²-7 t³-1 7 tan-¹t, Int Evaluate lim F(t).

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Q: Is there a vector field Ğ on R³ such that curl Ġ = (xyz,–y*z³,y*z*)?

A: the given function is, G→=xyzi^-y4z3j^+y3z4k^ If there exists a vector field G then the divergence…

Q: Prove that the divergence of a curl is always zero. Check it for function va = x2x-hat + 3xz2y-hat −…

A: Claim : (a) To prove that the divergence of a curl is always zero.               (b) Check it for…

Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: * Let X be the vector space of continuous real valued functions in [-1, 1]. Define (·, ·) : X×X → R…

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Q: Determine if the scalar vector field is Lipschitz continuous on the indicated domain ON = f(x) = tan…

A: The detailed solution is provided below.Explanation:Given, f(x)=tanx,Ω=(−2π​,2π​) f′(x)=sec2(x)…

Q: Determine if the scalar vector field is Lipschitz Continuous on the indicated domain f(x)=sin(tx), =…

A: The scalar vector field  f(t,x)=sin(tx) is Lipschitz continuous on the domain  Ω=R ,I=R with…

Q: Determine if the scalar vector field is Lipschitz continuous on the indicated domain f(x) = ex n = R

A: f(x)=ex is not lipschitz continuous in  Ω=RExplanation:Suppose f(x)=ex is Lipschitz on R.  Then,…

Q: Q1/Apply both ends of the theorem s'Stock if the vector field H is given by the relation H = а, at…

A: Given: By stoke's theorem  H = y2zxax + y2z22x2az    at the plane y=2 and the plane bounded by the…

Q: What is the divergence of the vector field F of F(x,y) = x² i + 2y²j?2 div F = 2x + 4y div F = 4x +…

A: Given: F→=x2i+2y2j

Q: 3. Determine if F = x*yi+ xyz j + x²y² k is a conservative vector field.

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Q: Determine whether or not the vector ficld F (r, y, z) = (2ry22, 3ry2, 2ryz) is conservative. If it…

A: The general form of a vector-valued function in three dimensions is given by F=F1,F2,F3, where F1,…

Q: Let P denote the set of all functions f on [0, 1] such that there exists at least one point x E [0,…

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Q: 5. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' = 4f}…

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