(a) Determine whether the field F(r, y, 2) = (-2, 2, -2az - 62) is conservative or not.

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

A:

Q: Find the wark done by the Foree Field FoK x>ye^> along the parabola X=y+1 from (bo) to (2,1).

A:

Q: Suppose the field lines of a gradient field F: R2 + R² are hyperbolas and the length of each vector…

A: To find- Suppose the field lines of a gradient field F : ℝ2 → ℝ2 are hyperbolas and the length of…

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: Q4 A) Calculate the divergence of the vector field F=y³i + xyj B) Find the Curl of F(xy.z)=3x²i +…

A: To find out the divergence of vector field.

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

A:

Q: 11) If f is a potential function for the vector field F = (-2y + 2xyz, -2x + x²z, x² y + 8z), then…

A: To find- If f is a potential function for the vector field F = -2y + 2xyz, -2x + x2z, x2y + 8z, then…

Q: find the divergence of the field. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)k

A: Given Data The field function is fx=x-y+zi+2x+y-zj+3x+2y-2zk.   The expression for the divergence of…

Q: Find the divergence of the field. F= (4x + 5y + 7z)i + (8x - 8y + 3z)j + (2x - 7y + 4z)k div F =

A:

Q: (2) For a differentiable vector field f, which of the following is (are) incorrect? (a) ▼ × (▼ × ƒ)…

A: Since we know Formula a→×(b→×c→)=(a→·c→)b→-(a→·b→)c→

Q: If f(x, y) is a potential function for the gradient vector field F(x, y) (Зх — 2у) і — (2л — у) ј, —…

A: Since F is conservativei.e, ∇×F  , so there exists some potential function f so that ∇f =F .

Q: For a force field F, if curl F= 0, then the field is nonconservative field. True False

A: False

Q: 3. Use the fact that F = is a conservative vector field to evaluate SF. dr when C is the path r(t)…

A:

Q: Find the divergence in (2,1,-1) for the vector field FCX₁Y₁² ) = x³y² 2 ² + x ² z 3 + x^²yk

A:

Q: Find the gradient field F = Vo for the potential function o = x²y-3y²x. F =

A: Given the potential function:φ=x2y-3y2xWe need to find the gradient field F=∇φ

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

A:

Q: Let f(x, y, z) = 3xyz + y?z. Find the gradient field Vf.

A: Formulae of gradient field is given below

Q: 3. Find a field F(x,y) = (M(x, y), N(x, y)) in the xy-plane with the property that at each point

A: Let f = xi + yj  Then F = ff = unit vector = xi+yjx2+y2

Q: the potential function f for the field F = 5x i + 9y j + 4z k. eral expression for the infinitely…

A:

Q: find the divergence of the field. 1. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)k 2. F = (x ln…

A: Consider the vector field Substitute the value of vector fields then Hence, the divergence of…

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

A:

Q: Consider the vector field F = ( 'x²+y2 'x²+y2/ Write yes or no for each: Is this field conservative…

A: The solution are next step

Q: The slope field oY = f(t,y), with initial condition f(0) = 1, is shown. Find limf(t.y). dt - 10 「 ンン…

A: On the slope field draw the graphical solution with initial condition f(0)=1:

Q: Find the gradient field F = Vq for the potential function o = 3x y-y²x. %3D F = O0

A:

Q: Find the work done by the three-dimensional inverse-square field F(r) : 1 r on a particle that moves…

A:

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

A:

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

A:

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

A:

Q: Use the method of isoclines to sketch the slope field for y =x +y-1, then draw the solutions…

A: In this question, we sketch the direction field for the differential equation y'=y+x2-1 To draw the…

Q: Determine whether the vector field is conservative. F(x, y) xex²2y(2yi + xj) conservative O not…

A:

Q: Which of the following graphs is the slope field for 15/1 1~/1ST ~\~ in 13t-ii-i ~~~ -1 1 2 3 4 5…

A:

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.  => x > 0 and y > 0 => x2 > 0 and - y < 0…

Q: Calculate ∇φ and ∇2φ of the scalar field: φ = e √ x^2 + y^2 + z^2

A:

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: (a) F = (10xz + y) i + 2xy} + 5x² k: curl F = F is a gradient field (b) G = (5xy + yz) i + (5x² + z²…

A: F→=10xz+y2i→+2xyj→+5x2k→ curl F→=i→j→k→∂∂x∂∂y∂∂z10xz+y22xy5x2 therefore F is a gradient Field

Q: 7. Choose an option in which the vector field is conservative: None of them OF = (xy, x+ y) O 2,…

A: A vector field F=∇f is conservative if and only if ∂M∂y=∂N∂x

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

A:

Q: Find the divergence of the field F. yj - xk F= 1/2 +: 2 OA. 3/2 2 -X O B. 3/2 2 X -y OC. 3/2 O D. 0

A:

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

A:

Q: Q5: a) Show that the vector field F = (x² - y² + x)i - (2xy + y)j, is irrational.

A: Given vector field is F→=x2-y2+xi^-2xy+yj^. To show: Vector field F→=x2-y2+xi^-2xy+yj^ is…

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

Q: If the scalar field is given by f = 2p², the result of V × Vƒ at P(4, ,4) 3

A: The expression for the gradient in the cylindrical coordinates  is given as . Following are the dot…

Q: 2 2 Find the gradient field of the function f(x.y.z) = In /x ?. + 4y +z The gradient field is Vf =…

A:

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

A:

Q: Let F be a field of characteristic not equal to 2. Let a, b E F such that b is not T CL If 1

A:

Q: ermine whether the vector field F(x, y, z) = (x³, 3zy², y³) is conservative on R³. If so, find a…

A:

Q: Q4: A- Find the flux of the field F = i+j+k √√x² + y² +2² bounded below by the plane z-0 and bounded…

A: