23.3k views
0 votes
Determine whether or not F is a conservative vector field. If it is, find a function f such that F = -f. If it is not, enter NONE. F(x, y) = (2x - 4y) i + (-4x + 10y - 5) j

f(x, y) = ___-______+ k

User JMRC
by
7.8k points

1 Answer

2 votes

We need to have


(\partial f)/(\partial x)=2x-4y


(\partial f)/(\partial y)=-4x+10y-5

Integrate the first PDE with respect to
x:


\displaystyle\int(\partial f)/(\partial x)\,\mathrm dx=\int(2x-4y)\,\mathrm dx\implies f(x,y)=x^2-4xy+g(y)

Differentiate with respect to
y to get


(\partial f)/(\partial y)=-4x+10y-5=-4x+(\mathrm dg)/(\mathrm dy)\implies(\mathrm dg)/(\mathrm dy)=10y-5


\implies g(y)=5y^2-5y+C

So we have


f(x,y)=x^2-4xy+5y^2-5y+C

which means
F is indeed conservative.

User Peteallen
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.