1. Let F(x, y) = (y, 2x2). Use the circulation form of Green's theorem to calculate the circulation f F·T ds, where C is the square with vertices (0,0), (1,0), (1,1), (0, 1), oriented positively.

Q: B) Evaluate the line integral F-dr, where F (x, y) = r*y°i+x²y°j and C is the curve r (t) = (t³ +…

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Q: Let f be a function defined in R2 by f(x,y) = 3x2 - 2x3 - 3y2 + 6xy.  Find the gradient vector in…

A: We have to solve given problem:

Q: Let C be the parabola y = x² for 0 < x < 2. Let F(x, y) = ye"i + xe*"j.Compute the line integral

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Q: Find the gradient field that corresponds to function #(x,y,z) = y² – x² + x e=² – y³ z

A: The gradient of a scalar point function f(x,y,z) is given by; Grad (f)=∇f=∂f∂xi+∂f∂yj+∂f∂zk

Q: Find the function f if. Vf-(3a²y + y², a² + 2xy), Vf dr with and without the Fundamental Theorem of…

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Q: Find the tangent plane for the level surface f(x,y,z) = 5x4 + cos (4y) + In (6z) = c at the point…

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Q: Demonstrate that the functions Y1 = a", y2 = sin(x), Y3 = xe* are linearly independent on R. 9.

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Q: Vy: y(0) = 1 dx dy Vy: y(0) = 0 %3D %3D dx dy - x-y; y(2) = 2 %3D dx dy VX-y; y(2) = 1 %3D dx

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Q: If F= represents the flow of a liquid through a parabolic cylinder S: y=x, 0sxs2, 0szsl. That is…

A: If F→=y2,-x2,z2 represents the flow of a liquid through a parabolic cylinder S:y=x2,0≤x≤2,0≤z≤1.…

Q: If F = xi + -4yj +x2k determine the flux of F through S, where S is the portion of z2 = x2 + y2,…

A: Given - F = x i^ - 4y j^ + x2 k^  ;             S is the portion of z2 = x2 +y2 , 1 ≤ z ≤ 3 To find…

Q: Study the analytically of w = L(2) at the points z = 0, z= v2,z = -21

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Q: (a) Find the solution of the limit using polar coordinate system. x² + y2 lim (x,y)-(0,0) x2 + y?…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, yo, 20) is given by the…

A: Given: Surface f(x, y, z)=c and a point x0, y0, z0 To show: Equation of tangent line to f(x, y, z)=c…

Q: Find the flux of F = yi + 2zxj – z²k through the parabolic cylinder y = x2,0 < x < 1,0 < z< 4, in…

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Q: Find the gradient field of the function, g(x,y,z) = e 3z - In (x +2y).

A: Given function  gx,y,z=e3z-lnx2+2y2                     (1) The gradient field is  ∇g=∂g∂x,∂g∂y,∂g∂z…

Q: Use the divergence theorem to find the flux of F = [x²yz, -xy² z, 6 – 5z2| on the hemisphere S =…

A: Consider the given function. The divergence theorem is defined as. Now, find the value of div F.

Q: find the outward flux of F across the boundary of D.  F = 2xyi + 2yzj + 2xzk D: The cube cut from…

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Q: Let F = (2z, - x, 3y): Find the flux of F, that is, - dS, where S is the graph of the function z =…

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Q: Consider the force field F = (ev – e2" , (x + })ev) and the curve C = C¡U C2 where Cı is the line…

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: Show that the function f(x, y, z) = x² + y² – 222 sat isfies the Laplace equation. - Good Luck.

A: Given function f(x,y,z)=x2+y2-2z2. We have to show function satisfies the Laplace equation. A…

Q: Q2. If F = (x² − y² + 2xz)ī − (2xz − xy + yz)j + (z² + x²)k, find the of curl F and curl of curl F…

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Q: 1. Evaluate the line integral f(z, y) ds where f(x, y) = Vx+ 2y and C is the straight-line segment…

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Q: Let f(x, y) = 8x + 2y2, r(t) = (cos(t), sin(t)). Use the Chain Rule to calculate f(r(t)). !! dt…

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Q: Show that the function f(x, y, z) = x² + y? – 2z² satisfies the Laplace equation.

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Q: Let F1 = (x+ y)Âx + (-x+y) ŷ– 2zî and F2 = 2 yx+ (2x +3z) ŷ +3yî. Calculate the curl and divergence…

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Q: Calculate the gradient, the divergence and the curl for the following fields, when possible: ₁ = e/y…

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Q: 4. Let f(x, y) = sin(x²y) + vx + e+y. Find the equation of the tangent plane to the surface z f(x,…

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Q: Find the circulation of F over the counterclockwise closed curve C: F= tan-1yi+ + 2x + y+z)j +yk…

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Q: Find the gradient fields of the functions ƒ(x, y, z) = ln√(x2 + y2 + z2)

A: The given function is fx,y,z=lnx2+y2+z2   The gradient field of a function is represented by,…

Q: Consider h(x, Y, z) = cos (xy) + eY% + ln (xz). Determine the directional derivative of h at the…

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Q: 2x Find the gradient vector field (F(, y, z)) of f(x, y, z) = y cos F(2,y, 2) =

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Q: Q. A-Find the gradient (Vf) for the following level surface at the given point: f(x, y, = x2 + y² -…

A: Given: f(x,y,z)= x2+y2-2z2+zlnx  Gradient,∇f=i∂f∂x+j∂f∂y+k∂f∂z

Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: Letf(x,y,z) = x^2+y^2+z^2.  Calculate the gradient of f.  Calculate ∫_C(Fdr)  where F(x,y,z)=(x,y,z)…

A: Given  f(x,y,z) =x2+y2+z2

Q: Find the tangent plane for the level surface f(x,y,z) = 7x5 + cos (5y) + In (9z) = c at the point…

A: Answer: In oerder to use gradients we introduce a new variable c= 7x5+cos(5y)+ln(9z) Therefore the…

Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

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Q: Suppose we have a surface f (x,y)=x²y+cos (4x) · Which of the following is the gradient of f(x,y) at…

A: First I have found the partial derivatives of f(x,y)  with respect to 'x' and 'y' at point P(π/2,…

Q: Let f = (r +2ryz)i+( + 2xy)j+ (+ 2xyz)k. Find j at (1.0, 1). (a) divergence of

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Q: 2. (a) Let z ry, r =r cost and y=r sin t. Compute by using the chain rule and express your answer in…

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Q: Evaluate the line integral y² dx + x²y dy where C' is the rectangle with vertices (0,0), (5,0), (5,…

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Q: Find the gradient field that corresponds to function (x, y, 2) y x +x e-y z

A: The given function ϕx,y,z=y2-x2+xez2-y3z The gradient of a function is given by ∇f=∂f∂x,∂f∂y,∂f∂z

Q: Find the direction of maximum change of f(x, y, z) = xey + z² at (1, In 2,2).

A: Introduction: A function's maximum directional derivatives at a particular point P are produced in…

Q: 4. Calculate the flux of F = over the square shown below, oriented counterclockwise. SS (Px +Q, R…

A: We have to check given work on this problem write or wrong   But this work is wrong Because flux or…

Q: Consider the following. Z = =DVx²+ y?, 5x - 2y + 3z = 22, (3, 4, 5) (a) Find symmetric equations of…

A: a) Here, f(x,y,z)≡x2+y2-z2=0g(x,y,z)≡5x-2y+3z=22P≡(3,4,5)

Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

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Q: Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following…

A: To determine the rate at which the box’s (a) Volume, (b) Surface area and (c) diagonal length are…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

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Q: Given that u = x? - y? and v = - 2xy, determine the stream function and potential function for the…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let if (x, y, z) # (0,0, 0) if (x, y, z) = (0, 0, 0) xyz x²+y²+z² > f(x, y, z) = { 10, (a) Use the…

A: fx, y , z = xyzx2 + y2 + z2, if (x, y , z) ≠(0,0,0)                             0,…