Answers

1. (a) We are given –

   F=(14x+8y)i+(8x+18y)j,

   so M(x,y)=14x+8y and N(x,y)=8x+18y.

   Therefore My = Nx = 18, and the vector field is conservative.

   (b) To find a potential function, we integrate

   M and N:

   ∫(14x+8y)dx=7x^2+8xy+f(x)

   ∫(8x+18y)dy=8xy+9y^2+g(x)

   Combining the results, and eliminating the duplicate terms, we get

   f(x,y) = 7x^2+8xy+9y^2.

   (c) The endpoints of C are (0,−1) and (1,0).

   Therefore the Fundamental Theorem of line integrals gives

   ∫c F⋅dr=f(1,0)−f(0,−1) = −2.

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   Disclaimer : The complete question is given below.

   Question :

   (a) Show that the vector field F = (14x+8y)i + (8x+18y)j is conservative.

   (b) Find a potential function f such that

   F = ∇f

   (c) Use the potential to evaluate the line ∫c F⋅dr,

   where c is the part of the unit circle in the 4th quadrant.

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