Let R be the region, 4 ≤ x² + y² ≤ 25 with the positively oriented boundary R. Let f(x, y) be a smooth real- valued function in R. (a) Find the value of [ (Zi+ Zj) · år aR (b) Let F(x, y) be a smooth vector field on R such that curl F = Find SOR F dr, using Green's theorem. aR √√x² + y²k₁

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

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A: Given  that   be a closed  curve  and  be  differtentiable functions.Now,recall  Stoke's  theorem :…

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A: Given:The vector field  is given by .To Find:We have to sketch the given vector field.

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A: The given function .We have to find the .

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A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

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A: We have to show .

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A:

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A:

Q: 3. Let F and G be vector fields with differentiable components. Express curl (F x G) in term of div…

A: Introduction: The curl of a vector is a vector-valued function. It is the cross-product of the…

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A:

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A:

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A: Since the given vector field F is shown in the xy-plane (2 dimensional).The vector field of the form…

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A: The solution is given as

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A: The given vector field is:F(x, y)=x-yi+x2+y2j