5. Let E be a dosed subspuce of E. Show that 11 cn3 |l = inf Il x-mll de fines E/M. a normed space ond MCE be %3D MEM on the quotient vector norm space

Q: Let F be the following vector field: F(x,y,z) =. A. Find div F B. Show that there is no vector field…

A: Given: Fx, y, z=e-x siny, e-ysinz, e-zsinx (A) To find: Div F. (B) To Show there is no vector G such…

Q: What is the 4-point DFT of x = (0, 1, 0, 1) ? O(1, 1, 1, 1) O (2, - i- 1, 0, i-1) O (2, 0, - 2, 0)…

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Q: Let x₁ = a +bt² +c+z₁, t=0, 1, 3,... where a,b and c constants and z, ~WN (0,0²). a) Let U₁ = x,…

A: Given, xt=a+bt2+c+zt , t=0, 1, 3, . . . where a, b and c constants and zt~WN0, σz2.

Q: T X = R with standard topology { IneN} CR k= Show that the subspace topology on k is discrete.

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Q: Let S = {A2«2| trace(A) = 0}, i.e. S = Show that -a S is a vector space.

A: The solution is given as

Q: 15. Let U and V be subspaces of a vector space W. Show that if W = UV, then UnV = {0}.

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Q: let X~beta(a,b). find m.g.f ofY=ln| 1-x

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Q: show thal the set of all poiuts in the Subspace of IR 3. plane ax +by+CZ =0 is +cz = ois CA

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Q: Find the orthogonal projection y of y = -{-----} = նշ W = Span ŷ Ex: 1.23 2 6 onto the subspace

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Q: Is the function foo= 1 %3D Integrable over the interval [o,2]? Justify

A: We will show that the given function,                   integrable over the interval.

Q: . In each case, the eigenvalues of a transition matrix A are given, as well as a basis for each…

A: (a) λ1=1 with basis 23 for Eλ1 and λ2=0.6 with basis 12 for Eλ2  ⇒λ1=1  and  λ2=0.6 so, stabilize…

Q: Let 10yx⁹ F(x, y, z) = (9x8 ln(5y² + 3) + 3z¹0)i + (- +4z)j + (30xzº + 4y – 4ñ sin лz) k 5y² +3 and…

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Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U + W.…

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Q: 1. Let S, T, T' denote three subspaces of a finite-dimensional space V for which the following…

A: Lets solve both questions

Q: In the metric

A: Given, In the metric subspace ℤ+, d of metric space ℝ+, d, where  dm, n=1m-1n; m, n∈ℤ+

Q: Let f(x) = cx^3.50. Suppose that f(y) is 2 times as large as f(z). How do y and z compare?

A: Given,           Let f(x)=cx3.50.Suppose that f(y) is 2 times as large as f(z).

Q: Find the orthogonal projection ŷ of y = onto the subspace W = Span { uị u2 5 Ex: 1.23 : ŷ =

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Q: , dim W 5. Let U, W be subspaces of a vector space V. If dim U and Un W = {0}, what is dim (U+ W)?

A: Let V be a vector space over a field k and let U, W be finite-dimensional subspaces of V.  Then…

Q: rcise 1.10. (a). Let V = R² = {(x.y)l x, y > 0} with usual addition and scalar multiplication. Is it…

A: Yes, the given set with usual addition and scalar multiplication is a vector space. Let's prove that…

Q: consider the subspace W of D given by W = span(e^2x, e^-2x) show that the differential operator D…

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Q: Let x be a vector in R^n (written as a column), and define U = {Ax | A ∈ Mmn}. Show that U is a…

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Q: Verify the Divergonce Theorem for F=(2x-x)+x*y} -aż? taken over the bounded by x -o, X=1, y=0,…

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Q: of f. Sketch the gradient field for f (x, y) = /x² + y? along with several level curves

A: Here we have,

Q: Is S = {(x1, x2)^T ∈ R^2| x1 > x2} a subspace of R^2? Justify your answer.

A: Solution:Given: we have to check that is a subspace of or not.

Q: Find Laplace operator for functier F = 2x' y? zZ %3D

A: The given function is F=2x3y2z2To find its laplace operator

Q: Let w = e®(z+=) dy ^ dz+e" dz ^ dæ + e(z =) dæ A dy be a two-form on R³, and o : (IR>0)² → R³ be the…

A: Given: A two-form on ℝ3, ω=e8x+zdy∧dz+e-ydz∧dx+e6x-zdx∧dy A smooth function ϕ:ℝ>02→ℝ3, where…

Q: Let M be closed subspace of normed space X. Show that MX iff int(M)=

A: Given Data: Let us consider the given data, M be closed subspace of normed space X. To Show that:…

Q: V = M2x2 (R) let M = { A€ V A = A²} N = { A = V A = -A ² } Subset of the vector space V V = MON…

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Q: Let X1, , X, be iid with population pdf given by h ( which integrates to 1 over the real line. Here,…

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Q: Describe the different methods available for evaluating the integral of a conservative vector field…

A: To Describe: Different methods available for evaluating the integral of a conservative vecctor field…

Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 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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Q: If B' CB is a subspace, and we let E' = p-1¹(B'), show that p: E'→B' is a covering map.

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Q: approximationto fx)=coshix using legendrepolynomials Find aquadratic least-squares approximati onto…

A: Using Legendre polynomials, We have to find a quadratic least-squares approximation f(x) = cosh(x).

Q: Let Find o*w. w = e e6(x+²) dy ^ dz + e¯ª dz^ dx + e¹(x−²) dx / dy be a two-form on R³, and 6:…

A: Given: A two-form on ℝ3; ω=e6x+zdy∧dz+e-ydz∧dx+e4x-zdx∧dy. A smooth function, ϕ:ℝ>02→ℝ3 defined…

Q: Prove that if f: ℝ^n → ℝ^m is differentiable at a ∈ ℝ^n, then f is continuous at a. Hint: Apply a…

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Q: Pind faurier Tran h-

A: According to our guidelines, we are supposed to answer only one question. (1)  In this question, we…

Q: 7. Imagine 'gluing' together two tetrahedra with a common interior face that is removed. The…

A: We have to show that the sum of the outward pointing area vectors of the resulting polyhedron with 6…

Q: d (n, m): = 0 si n = m si n‡m 5-c(n-m)

A: Question seems wrong

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: