(7) Let F(x, y, z)= y cos(ry) i+r cos(ry) j-sin(z) k. (a) Find curl F. (b) Is the field F conservative? (c) If F is conservative, then find its potential.

Q: 5. Consider the vector field F(x,y,z) = yzi + xzj+(xy+2z) k and the curve C given by r(t) = (t+ sin…

A: Given vector field: Fx,y,z=yzi+xzj+xy+2zk Curve C is given by: rt=t+sin12πt,t+cos12πt,2t,   0≤t≤1…

Q: 1. Let F(x, y, z) be the vector field (x² (1 + y²z²), Q(x, y, z), ey cos(22)), where is a…

A:

Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

A:

Q: !

A: We have to find

Q: Is the vector field F(x, y) = (In y + sin(e²“), #

A:

Q: (1) Find a potential function for the vector field F given by F(x,y) = (5x-2y)i - (2x - e³y)j, (x,y)…

A:

Q: Consider the vector field 1 G(x, y, y, z) Cos y sin y 2√x' xsin y' 1+z² Show that G is conservative…

A:

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: Find a potential field for F(x, y, z) = (y +z)i + (x + z²)j + (x+2yz)k or prove none exists.

A:

Q: 6. Consider the vector field defined by y F = (²) ² + ( 3y² m (x) - ₂ √ √ ) ³ + ( √T)k i j (a) Use…

A:

Q: (a) Find a potential field function for F(x, y) = (x + y)i + (x - y) or prove none exists. (b) Find…

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Q: (i) (ii) State the three physical interpretations of the divergence with a sketch of their vector…

A:

Q: 8. The component of the field F= (1,2,3) normal to the plane z = -3x + 3y -1 is (a) 0. (b) –2//19.…

A: If we are given a plane equation in the form Ax+By+Cz=D, then the normal vector to the plane will…

Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: (a) Show that the vector field F (x, y, z) = (x² + y², x, z³) is not conservative.

A:

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: Briefly describe how to find a potential function φ for a conservativevector field F = ⟨ƒ, g⟩ .

A: Before we get into the procedure to find the potential function, we need to understand certain…

Q: Find the potential function f for the field F. F = (y - z) i + (x + 2y-z)j- (x+ y) k O f(x, y, z) =…

A: We have to find

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: The vector field V(r) = v(p) where p = √² + y², and is the unit vector of polar coordinate. √/z²+y²…

A:

Q: Find the work done by the force field F(x, y) = x sin(y) i + yj on a particle that moves along the…

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Q: Find the potential function f for the field F. F = (y - z) i + (x + 2y - z) j - (x + y) k   a)…

A: Let us find the potential function

Q: 1. Determine whether F(x, y) et sin yi + e" cos syj is a conservative vector field.

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Q: Given the vector field F(x, y, z) =(z cos(xz) + ze“ )î +(3y² +1)j+(xcos(xz) + xe )k 4. (a) Find the…

A: Given F→x,y,z=z cosxz+zexzi^+3y2+1j^+x cosxz+xexzk^

Q: ) Let F(x, y) = (3x² + y²)i + 2xyj. (a) Test if F is a conservative vector field. (b) If F is…

A:

Q: (a) Let F = 2xz²i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii)…

A: As per our guidelines we can answer only 1 question and 3 subparts of a given question so kindly…

Q: y +3z x + 2z 2y + 3x Consider the conservative vector field F(x, y, z) = with domain R³. Find a…

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Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

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Q: Let F be a vector field and Ca curve with parametrization 7 (t) where a <t<b. F. dr = | (cos(t³),…

A: