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-BARSIC- [3]
3 years ago
9

PATH INDEPENDENCE? 13-19 from Check, and if independent, integrate from (0, 0, 0) to (a, b, c) 13. 2e (x cos 2y dx - sin 2y dy)

Mathematics
1 answer:
TEA [102]3 years ago
4 0

The integral is path-independent if there is a scalar function f whose gradient is

\nabla f=(2e^x\cos2y,-\sin2y)

(at least, that's what it looks like the given integrand is)

Then

\dfrac{\partial f}{\partial x}=2e^x\cos 2y\implies f(x,y)=2e^x\cos2y+g(y)

Differentiating both sides with respect to y gives

\dfrac{\partial f}{\partial y}=-4e^x\sin 2y\neq-\sin2y

so the line integral *is* dependent on the path. (again, assuming what I've written above actually reflects what the question is asking)

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