Question

for each of the following vector fields, f, determine whether or not it is conservative. for each conservative field, find a potential function f that satisfies ∇f = f and f(0,0) = 0. if it is not conservative, type n.
a. f(x,y) = (-10x + 7y)i + (7x + 2y)j
f(x,y) =

b. f(x,y) = -5yi - 4xj
f(x,y) =

c. f(x,y) = (-5 sin y)i + (14y - 5x cos y)j
f(x,y) =

note: entered answers should be either expressions of x and y or the letter
\.

Explanation:

For Vector Field A

Step1: Check conservativeness

Let $\mathbf{F}(x,y) = P\mathbf{i} + Q\mathbf{j} = (-10x + 7y)\mathbf{i} + (7x + 2y)\mathbf{j}$. Compute $\frac{\partial P}{\partial y} = 7$ and $\frac{\partial Q}{\partial x} = 7$. Since $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, it is conservative.

Step2: Find potential function

Integrate $P$ with respect to $x$: $f(x,y) = \int (-10x + 7y)dx = -5x^2 + 7xy + g(y)$.
Differentiate with respect to $y$: $\frac{\partial f}{\partial y} = 7x + g'(y) = Q = 7x + 2y$. So $g'(y) = 2y$, integrate to get $g(y) = y^2 + C$.
Use $f(0,0)=0$: $0 = -5(0)^2 +7(0)(0) + 0^2 + C \Rightarrow C=0$. Thus $f(x,y)=-5x^2 +7xy + y^2$.

Step1: Check conservativeness

Let $\mathbf{F}(x,y) = P\mathbf{i} + Q\mathbf{j} = -5y\mathbf{i} -4x\mathbf{j}$. Compute $\frac{\partial P}{\partial y} = -5$ and $\frac{\partial Q}{\partial x} = -4$. Since $\frac{\partial P}{\partial y}
eq \frac{\partial Q}{\partial x}$, it is not conservative.

Step1: Check conservativeness

Let $\mathbf{F}(x,y) = P\mathbf{i} + Q\mathbf{j} = (-5\sin y)\mathbf{i} + (14y -5x\cos y)\mathbf{j}$. Compute $\frac{\partial P}{\partial y} = -5\cos y$ and $\frac{\partial Q}{\partial x} = -5\cos y$. Since $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, it is conservative.

Step2: Find potential function

Integrate $P$ with respect to $x$: $f(x,y) = \int (-5\sin y)dx = -5x\sin y + g(y)$.
Differentiate with respect to $y$: $\frac{\partial f}{\partial y} = -5x\cos y + g'(y) = Q = 14y -5x\cos y$. So $g'(y)=14y$, integrate to get $g(y)=7y^2 + C$.
Use $f(0,0)=0$: $0 = -5(0)\sin 0 +7(0)^2 + C \Rightarrow C=0$. Thus $f(x,y)=-5x\sin y +7y^2$.

Answer:

$-5x^2 + 7xy + y^2$

For Vector Field B