Consider the vector field F = (-y,x). Check if the vector field is conservative. Consider the line integral F dr, where C is the counterclockwise (positively oriented) circle of radius 3 centered at the origin. i. Calculate the line integral using the regular line integral method, using either SF(t) r'(t) dt or ſ. P dx + Q dy. ii. Calculate the line integral using Green's Theorem.

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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

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Q: Find (a) the divergence of the vector field and (b) the curl of the vector field.

A: Given :   To find : a) The divergence of the vector field. b) The curl of the vector field.

Q: Consider vector field: F = e i-cos y j+ sin² z k. Obtain divergence and curl for F at points (0,,).

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Q: Consider the vector field F(x, y,z)= 2.xy i+(x² +z² ) j+2yzk. c) Is F a conservative vector field?…

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Q: Calculate the divergence of the vector field. F(x, y) = 3x°i – 2x sin(xy)j div(F) =

A: Given vector function F(x,y)=3X3 i -2x sin(xy) j divF=i∂∂x+j∂∂y+k∂∂z.F

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Q: Suppose the vector field F = yi + (x + Z)j + (y + 2z)k is conservative. Find a potential function of…

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Q: Find the conservative vector field for the potential function h(x,y,z)=3xyln(x+y) by finding its…

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Q: Consider the vector field F = (7x + 3y, 3x + 2y) Is this vector field Conservative? ( Conservative…

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Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

Q: Evaluate the line integral of the vector field F(x, y, z) = (x²y³, e™y+z, x+ z²) around the circle…

A: Solution is as below

Q: Consider the vector field: F(x, y, z) Show that F is not Conservative, i.e., not a gradient. sin…

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Q: Determine whether the vector field h(x,y) = y^2i + (2xy-e^2y)j is conservative or not. If it is…

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Q: Given a vector field F = (x + xy?)i + (5xy)j. Use Green's Theorem to evaluate F- di where C is a…

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Q: Determine whether F(x, y) = xi +9yj is a conservative vector field. If so, find a potential function…

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Q: Consider the following vector field F (x, y, z) = yz cos (xy) i+ xz cos (xy) j+ sin (xy)k (a) Use…

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) = x²…

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Q: Determine whether the vector field is conservative. If it is, find a potential function for the…

A: Vector field F is conservative when curl F=0 Consider  F(x, y)=-yx2i+1xjThencurl F=∇×F…

Q: Consider the vector field F = (x² + y², 7xy). Compute the line integrals f. F· dr and f F· dr, where…

A: Given F=x2+y2,7xy, c1t=t,t2 and c2t=t,t for 0≤t≤1. We know that ∫cF·dr=∫abFxt,yt·C'(t)dt. When…

Q: Please compute the work done by the vector field F(x, y) = (sin x+xy²)i+(e" + a2)j in R?, where C is…

A: ..

Q: Consider the vector field F = a?i+ ?j+ y²k Calculate the divergence: Calculate the curl, Suppose S…

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Q: Show that the vector field given below is conservative and find its potential function. F(x, y) =…

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Q: The vector field F=(x+2y)i+(2x+y)j is conservative. Find a scalar potential f and evaluate the line…

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Q: 6. Consider the vector field F = (-y,x). a. Check if the vector field is conservative. b. Consider…

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Q: For the vector field F = (1/z)I -4j -(x/z^2)k , find the potential function f for the field.

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Q: For vector field F(x, y, z)=(1+ 9xry, 3x +e, ye' +2= (a) Carefully calculate curl F. (b) Find the…

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A: The answer is given by using the formula of line integral

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Q: The vector field F = 2e^(y)i + (2xe^(y) + y)j is given. a-)Determine whether the vector field F is…

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Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

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Q: Use the divergence of the vector field to compute F(x, y) = (tan-1 (2²)) i + (xy-2 +82²) j 4 fo…

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Q: Compute the line integral of the vector field F = (4y, –4r) over the circle z2 + y? = 16 oriented…

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Q: F=cos(x) î+sin(y) ĵ+cos (z) k

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Q: Consider the vector field F = (4x + 5y, 5x + 7y) Is this vector field Conservative? Conservative If…

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Q: Consider the vector field F = (x*y, x*y? Is this vector field conservative? Select an answer If so:…

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Q: Determine whether the vector field is conservative, and, if so, find a potential function. (If the…

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Q: Let D be the upper half-plane (y > 0) and consider the vector field F : D+R² given by F(x, y) = (…

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Q: Determine whether F(x, y) = e5 cos 5y i – e5" sin 5yj is a conservative vector field. If so, find a…

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Q: Determine whether the vector field is conservative, and, if it is, find the potential function. (If…

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Q: Let a vector field F(x,y)=sinx i + xy j be given. What is the integral representing the work done by…

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Q: For the vector field F(x, y, z) = 1. a) Find curlE b) Find div E c) State whether or not the vector…

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Q: Evaluate the line integral of the vector field F = xy²i+ x²yj along the curve r(t) = (t+sin,t+cos-…

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