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devlian [24]
3 years ago
6

Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. (if the vector field is

not conservative, enter dne.) f(x, y) = (2x − 6y) i + (−6x + 10y − 9) j\
Mathematics
1 answer:
gladu [14]3 years ago
3 0

\mathbf f is conservative if we can find a scalar function f such that \nabla f=\mathbf f. This is equivalent to solving the system of PDEs,

\dfrac{\partial f}{\partial x}=2x-6y

\dfrac{\partial f}{\partial y}=-6x+10y-9

Integrate both sides of the first PDF with respect to x:

f(x,y)=x^2-6xy+g(y)

where g is some function independent of x. Then differentiatng with respect to y, we have

\dfrac{\partial f}{\partial y}=-6x+\dfrac{\mathrm dg}{\mathrm dy}=-6x+10y-9

\implies\dfrac{\mathrm dg}{\mathrm dy}=10y-9\implies g(y)=5y^2-9y+C

and so \mathbf f is indeed conservative, with

f(x,y)=x^2-6xy+5y^2-9y+C

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