Consider the optimisation of the scalar field f(z,y) = (y+)(y² + 3y + z²) on the set D= {(z,y) | z² + y < 9}. Calculate the eight stationary points of f on D. You do not need to determine the character of these stationary

Q: Find the divergence of the vector field F = ā x7, if r = xi+yj+ zk and ā = 3 i+3j+9k, by first…

A: First of all we calculated the vector F by calculating the vector product of the vectors a and r.

Q: The vector field F = z tan-(y²) i + z³ In(x² + 1) j + (cos z + ay) k has a vector potential with…

A: The solution are next step is

Q: 4- Verify the divergence theorem for the vector field Á = 3p² â,+2p sin(o) âs+2k for the cylinder…

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Q: Find the divergence of the vector field.

A: To Find the divergence of the vector field.

Q: 1. Consider the vector field F(r, y) = -y²i+(x+2)j. (a) Evaluate F(0, 1). F(0,1) = (b) Sketch the…

A: Given:

Q: Find the potential function f for the field F. F= (4x3y10z10, 10x4y9z10 10x4y1029) O fix, y. 2) =…

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Q: Find the Laplacian of the scalar field C = e-* sin 0 cos & at the point (1, n/3, n/6)

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Q: Find the divergence of the vector field F. F(x, y) = 3xe*i - 9x²y²j

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Q: 2) For the vector field F(x, y, z)=(xz²-x²y)i + (xy²-y³2²)] +xy²z³k, find the following: a) Find the…

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Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

A: To use Green's theorem to find the counterclockwise circulation and outward flux for the field F and…

Q: Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field.…

A: We will use Integration and concept of total differentiation for finding scalar potential of vector…

Q: Prove that F(x, y, z) = (y² cos x + 2*)i+ (2y sin x – 4)j+ (3xz2 + 2)k - is a conservative vector…

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Q: Calculate the divergence of the vector field F=x²y³z¹i-xyzj+(x+y+z) k

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Q: Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

A: This question is belong to the vector calculus in which we have to deal with the divergence theorem…

Q: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

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Q: Find the flow of the field F = V(x²ze) once around the ellipse C in which the plane x + y + z = 1…

A: Consider the given field, F=∇x2zey around the ellipse in which the plane x+y+z=1intersects the…

Q: 4. (a) Show that the divergence theorem holds for the vector field F = 2xî + yz} – 3z²k where o is…

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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: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

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Q: Consider the scalar field f(x,y, z) = 2x² – y² + 3z? Determine V Vfl(1.-1.1) II. VxVfl1.-1.1)

A: Here, the given function is: f(x,y,z)=2x2-y2+3z2

Q: Find the divergence of the vector field F. F(x, y) = 4xe*i - 6x²y²j

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Q: Find the net outward flux of the fieldF = ⟨bz - cy, cx - az, ay - bx⟩ across any smooth closed…

A: The vector field is F=bz-cy,cx-az,ay-bx where a, b, and c are constants.

Q: Determine whether the following vector field is conservative on an open region R of R2 that does not…

A: For our guidelines answer only  question 2 but I give answer of both questions

Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

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Q: b) Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

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Q: Is the vector field F (x, y) = (2xy – y³) î + (x² – 3xy²) ĵ solenoidal, conservative, both or…

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

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Q: a) Determine whether or not the vector field F 8yz, – 6 cos(y) – 8xz, – 8xy + 10z > - is…

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Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 6. Find the circulation of the field F(x, y) = -yi + x²j along the closed elliptical path x² + = 1…

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Q: 1 Find the divergence of the vector field F -(xi + yj+ zk) Vx? +y? +Z 2 x² + y² + z? 1 x² + y? + z?…

A: If the vector field is given by F=F1i+F2j+F3k then the divergence of F is ∇·F=∂F1∂x+∂F2∂y+∂F3∂z.

Q: Find the divergence of the field. F = (- 6x + 8y – 4z)i + (- 8x – 5y + 3z)j+ (x + 5y + 2z)k div F =

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Q: 2x 2y Consider the vector field F = |x² + y2’ x² + y² (a) If C1 is the unit circle x² + y² = 1,…

A: Compute the circulation and flux using parameterization:  ∮C1Fdr=∮C12xx2+y2,2yx2+y2·dx,dy…

Q: Consider the optimisation of the scalar field f(r,y) = (y+2)(y² + 4y+z²) on the set D = {(x,y) | x²…

A: Given function is, fx,y=y+2y2+4y+x2 Differentiating the given function partially,…

Q: Find the divergence of the vector field F. F(x, y) = 4x²i + 5y²j

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Q: Find a potential function for the vector field F by inspection. (If an answer does not exist, enter…

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Q: Consider the vector field F(x, y) = (x², −y). If (x, y) is in the first quadrant then the vector…

A: Part 1. (x, y) is in the first quadrant.  =&gt; x &gt; 0 and y &gt; 0 =&gt; x2 &gt; 0 and - y &lt; 0…

Q: Calculate the flux of the field F=xi+yj across the closed plane curve C, where C is the closed…

A: theory: graph:

Q: (a) For any vector field F, the gradient of the divergence of F is always zero.

A: Since the gradient of divergence of F is given by

Q: Determine whether the vector field is conservative. F(x, y) = xe(x^2)3y(2yi + xj)   If it is, find…

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Q: Show that V x VG = 0 for all scalar field G(x,y,z). %3D

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Q: Find the divergence of the vector field F. F(x, y) = 5x2i + 5y²j

A: The given vector field is: F(x,y)=5x2i+5y2j Let, F(x,y)=F1i+F2j⇒F1=5x2, F2=5y2

Q: 13. Calculate the divergence of the vector field F = x1e1 + 2x1e2 + +nx1en. ...

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Q: 3. Use the divergence of the vector field to compute F(x, y) = (tan-1 (2x)) ₁ + (xy-2 +²8²) J j 4 fo…

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Q: Find the divergence of the vector field F. F(x, y) = 4xe*i - 5x2y²j

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Q: Let C be the curve of intersection of z = 1 – 2x2 and the cylinder x2 + y2 = 1 oriented…

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Q: Find the potential function f for the field F = 6xi + 6yj + 4zk. %3D

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Q: Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and…

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Q: 1. For the scalar field 0 = x3 + 2xy in the domain ABC below, calculate the gradient field and then…

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