Given the bases B = {u1,u2} and B'={u'1, u'2} for R, where and u'1= 1 The transition mat rix PB B from B to B'is: 1 1 O A. 0-1 11 O B. 0 1 2 OC. -1 -1 -1 OD.
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Q: Use the Chain Rule to find Əz/əs and əz/ət. z = x°y', x= s cos(t), y = s sin(t) az as %3D az %3D at
A: Total derivative
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A: Given: x=set and y=tes.
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A: Introduction: By applying Green's theorem, one can determine line integral through surface integral.…
Q: Use the Chain Rule to find ôz/ðs and ôz/ôt. z = tan(u/v), u = 9s + 8t, v = 8s – 9t as az at
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Q: 1) Find the outward flux of F across the region D where F= (6x² + 2xy, 2y + x²z, 4x² y³ ) and the…
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A: Yes , we can solve using stokes theorem. Note that all the cases z = 0
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Q: Question 8 Given the bases B = {u,,u2} and B'= {u'1, u'z} for R², where and u' The transition mat…
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Q: Use the Chain Rule to find dz/ðs and əz/ət. z = tan(u/v), u = 5s + 8t, v = 8s – 5t %3D az %3D as az…
A: We have to use chain rule to find the partial derivatives of z.
Q: Find z dV, where E is the solid tetrahedron with vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,4)
A: x varies from 0 to 1& y varies from 0 to 2 and z varies from 0 to 4.
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Q: Consider the solid lih S,: X=4-y² そ=2
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Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D
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Q: Use the Chain Rule to find az/ds and az/ôt. z = tan(u/v), u = 3s + 6t, v = 6s - 3t az as az at II
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Q: 23. Give a formula that gives the length of the outer loop for the curve: r= 4-8 cos(0) (Hint: Graph…
A: r=4-8cos(θ)For loops crossedr=04-8cos(θ)=08cos(θ)=4cos(θ)=12θ=π3,5π3(Have symmetry from 0 to π and π…
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Q: Use the Chain Rule to find ôz/ds and ôz/ðt. z = tan(u/v), u = 5s + 9t, V = 9s – 5t az as az at
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Q: Given F(x, y, z) = (tan y + cosh x Joi 2xy)i + (x sec² y - x² + 1)ĵ - 2zk, evaluate F. dR where C'…
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Q: 1. The accompanying figure shows two polygonal paths in space joining the origin to the point (1, 1,…
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- Define T:R2R2 by T(v)=projuv Where u is a fixed vector in R2. Show that the eigenvalues of A the standard matrix of T are 0 and 1.Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.Let V be the vector space of functions which have B={sin ?, cos ?} as a basis, and let D be the differential operator on V. Find the characteristic polynomial Δ(t) of D.
- Determine the total work, by Green's theorem, done by the force field F(x, y) =< 2x - y2, x² y>, on a particle following the path described in the graph below.Find potential functions for the fields in Exercises 33 and 34. 33. F = 2i + (2y + z)j + (y + 1)k 34. F = (z cos xz)i + eyj + (x cos xz)kCompute the flux ∫F · n ds of F across the curve C for the given vector field and curve using the vector form of Green’sTheorem. F(x, y) =〈xy,x − y〉across the boundary of the square −1 ≤ x ≤ 1, −1 ≤ y ≤ 1
- Given the vector field F below (a) sketch its vector field over the region x∈[-1,1] and y∈[-1,1] (b) find its potential function if it exists. F(x,y,z)=<2xy+ex, x2>Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line integral ∫C F ⋅ dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = ⟨ƒ, g⟩ = ⟨2x - y, -x + 2y⟩ . Use the procedure given above to construct potential functions for thefollowing fields. F = ⟨x, y⟩Use Green’s Theorem to evaluate the work from the vector field F(x,y) = 〈-4x2+y2, 2x2+y〉along the path C shown in the picture below
- 5. Find a potential function corresponding to the vector field F(x, y, z) = (2x, 3Y, 4z).Alternative construction of potential functions in ℝ2 Assume the vector field F is conservative on ℝ2, so that the line integral ∫C F ⋅ dr is independent of path. Use the following procedure to construct a potential function w for the vector field F = ⟨ƒ, g⟩ = ⟨2x - y, -x + 2y⟩ .a. Let A be (0, 0) and let B be an arbitrary point (x, y). Define φ(x, y) to be the work required to move an object from A to B, where φ(A) = 0. Let C1 be the path from A to (x, 0) to B, and let C2 be the path from A to (0, y) to B. Draw a picture.b. Evaluate ∫C1 F ⋅ dr = ∫C1 ƒ dx + g dy and conclude thatφ(x, y) = x2 - xy + y2.c. Verify that the same potential function is obtained by evaluatingthe line integral over C2.Find the work that is done by the vector field F = x i + 2xy j + 3xyz k, on a particle that moves along the helix, r(t) = < cos(t), sin(t), t> from t = 0 to t = 2pi.