If F (x, y, z) = Vy (x, y, z), where o (x, y, z)s a scalar field, which of the following must be true? 1. V.F (x, y, z) = 0 OF, 2. ду az ƏF, 3. dy az Here the subscripts x, y, zindicate components of F. 1 and 2 2 only None of them All three of them

Q: 2. Don't generate a vector field- answer based on thinking about x and y values (pos, neg, zero....)…

A: In this question, we will find the direction of  vector field.

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: e work done between the points P, (0,0,0) and P2(1,1,1) for the conservative field F shown b 3y? -…

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Q: Vector fields V and W are defined by V- (2х — Зу + z, -3х — у + 4z,4y + z) W- (2x - 4y — 5z, - 4x +…

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

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Q: A vector field and a point in the vector field are given. Determine whether the point is a source, a…

A: We find the divergence at the point. If the divergence is positive , then the point is a source. If…

Q: Find a vector field F such that curl F = 8 i – 7j + 4 k. [Hint: Try F = T × 7 for some vector V ) °F…

A: Yes, there is a method to directly approach the answer with out just cross checking the options.

Q: What is the divergence of the vector field f = x²i+ xyzj – zk at the point (1, 2, 3). Select one: а.…

A: f=x2i+xyz j-zK at (1,2,3)  div f=i ∂f∂x+j.∂f∂y+K.∂f∂z

Q: A culture initially has P number of bacteria. At t = 3hour the number of bacteria is measured to be…

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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: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (x, , y.) +…

A: We have to show that the set V with the addition operation x1, y1+x2, y2=x13+x233, y13+y233 and…

Q: Determine which theorem applies to simplify the following. It is possible that more than one will…

A: Given  Find the work done by the vector field Fx,y,z=yzcosxyi^+xzcosxyj^+sinxyk^ along the…

Q: Let U and V be vector spaces over the field F, dimU=3, dimV=4. Which of the following pairs is not…

A: Step 1:  Step 2:  Step 3:

Q: Let C be the triangle with vertices. l6,0), (22,0), aed Ge,25) traveling Coaterdlehmle' let F be the…

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Q: If p (x, y, z)s a scalar field and A (x, y, z) a vector field, which of the following quantities…

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Q: 3) V. (FF) = f (V. F ) + F •V ƒ. 4) V: (F +G) = V. F + v•Ğ. 5) V: (FxG) = Ġ- (V × F ) – F · (V × Ġ)

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Q: Find the work done by the Öeld F = yzi + (xz + 3)j + xyk along the line joining the point A(1, 2, 0)…

A: Given field is,  F = yz i^ + xz+3 j^ + xy k^   The given points is A(1, 2, 0) and B(2, 0, 3). So,…

Q: (xy³ +3x²y, 3xy² + x²) 2+x ху у,

A: According to question given that Vector Field F⟨xy3+3x2y, 3xy2+x2⟩

Q: 9. If S and T are finite spanning sets for a vector space V, then S and I have the same number of…

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Q: HIf vectors x, y, z, u in the real vector space p (the polynomials) are defined by x(t)=1, y (t)=t,…

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Q: 13. Let V = xi+yj (a) Give reasons why V(x, y) is a vector field (b) Sketch the vector field V(x, y)

A: (a) Consider the given Vector, V=xi+yj The definition of vector field is given as " A vector field…

Q: Let B = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} be bases for…

A: a)b)c)d)

Q: Q4. A vector field F is defined as F = psin pa,+ 2p² zas + zcos oa,. The V × F at point P(1, 7/2, 2)…

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Q: 3. A finite-dimensional Hilbert space is a finite-dimensional complex vector space H together with…

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Q: b]= [13] -2 B= (b₁ b₂]=[ U=√fb₁ b₂ly and a= (-21, 47, 2, 4) Find a vector z & (Col B)t whose length…

A: We prove that  U is orthogonal basis of col(B):-

Q: Let s be a subset of a vector space y over a field F. Which of the following is true? A. If a- u ɛ s…

A: Given that V be a vector space over a field F.

Q: If f is a power function for the vector field F = (2xz, −3z, x2 −3y+12z2), then the value of f(−3,…

A: We have F=2xz, -3z, x2-3y+12z2 Now, F=∇f=∂f∂x, ∂f∂y, ∂f∂z=2xz, -3z, x2-3y+12z2 So, ∂f∂x=2xz…

Q: A vector field representing wind speeds is given by v = æy³ i + 3x³y³j. %3D A walker can take one of…

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Q: {{::) eR}. sider V = : #1, 22, 23 > 0 and ¤1, x2, x3 ER pose the vector addition and scalar…

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Q: If φ and ψ are smooth scalar fields, show that ∇×(φ∇ψ)=∇φ×∇ψ

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Q: Let F(x, y, z) = (yz², xz², 2xyz) and f be a scalar field such that Vf = F. Evaluate f-(1, –1, 1).…

A: We have F = (yz2 , xz2 , 2xyz)

Q: vectoral field. In each option there is a which option is garantzed to have an modependent traye…

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