Question

) Show that F is a Conservative Vector Field. (7b) Find the Potential Function f(x,y) for the Vector Field F. (7c) Evaluate W

1. kimchi2
(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.