Find a formula F = (F₁(x, y), F₂(x, y)) for the vector field in the plane that has the properties that F(0, 0) = (0,0) and that at any other point (a, b) ‡ (0, 0) the vector field Fis tangent to the circle x² + y² = a² + 6² and points in the counterclockwise direction with magnitude ||F(a,b)|| = 3√√a² + b².

Q: Find the work done by a force F = 6ỉ + 5j– 3k when a particle moves from point P(-3, 4, 5) to point…

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Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: x-y, z + y + 3,z²) and the net is decribed by the equation y = 1- x² - z on

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Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 3) to (9, 3), and the line segment…

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Q: Suppose f(x, y) = sin( (a) Vf(x, y) = = (b) Vƒ(3, π) = 2y x T 2y 2y 2y cos cos 2) Con (2) 200m (20 -…

A: Given that , fx,y=sin2yx and u is the unit vector in the direction of 1,1.

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: We have to find the rate of heat flow out of a sphere of radius .

Q: What is the work done by a force field F (x, y) := 2x to move a particle in a straight line from (0,…

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Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: Since the surface is whole sphere, that is S is closed so we can apply Gauss Divergence Theorem.

Q: Find the work done by the force field F(x, y) = 2xy³i+ 4x²y² in moving a particle counterclockwise…

A: The work done of a force F→ along a curve r→ is defined by ∫CF→⋅dr→.Since curve is closed in…

Q: Calculate the flux of the vector field F(x, y, z) = (4x+8) through a disk of radius 6 centered at…

A: Given,                    To find the flux of the given vector field through a disk of radius 6…

Q: For the vector field (x2,0₁ - 2x² -z²), evaluate its line integral along the closed curve, made by…

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Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

Q: The gradient of f(x, y) = x sin (y²) is the vector field F (x, y) = at (2,√π/6) and ending at…

A: Here we are given that the gradient of f(x,y)=xsin(y2) is the vector field F(x,y)=sin(y2)2xycos(y2)

Q: Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work…

A: Given point (2,4)  and (3,9)now equationof line passing through two point is given by…

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: We have to find the heat flow out of the sphere of radius

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: A sphere of radius 1 is x2 + y2 + z2 = 1. Parametrize the sphere as follows:Now,calculate the…

Q: find the work done by the force field F(x,y) =(2y+x²,x²-2x) acting on an object as it moves along…

A: Given: The force field, Fx,y=2y+x2,x2-2x.   To find: The work done by the given force field acting…

Q: A force field (in newtons) is defined by F(x, y) = [ + + x, 1 x² +1 y³ зав where coordinates x and y…

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Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

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Q: Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve that consists of two line…

A: Given that  Fx,y = xy2+2x+1 , x2y - 3y is a vector field. Let C be the curve that consists of two…

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

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Q: : (x² + y²)e-(x*+y -5), Find the rate of change of f at X = (1, –2) in the direction V pointing…

A: Given that fx,y=x2+y2e-x2+y2-5 The objective is to find the rate of change of f at X=1,-2 in the…

Q: Verify the Divergence Theorem for F = (-2, 3, 2³) across the surface of a sphere of radius 4 with…

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Q: Calculate the divergence and the curl of this vector function.

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

Q: For the vector field (x, y, z) evaluate the line integral along the curve made of three adjacent…

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