Question

For each of the following vector fields F , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, \\abla f = \mathbf{F} ). If it is not conservative, type N.

A. \mathbf{F} \left( x, y \right) = \left( -10 x + 7 y \right) \mathbf{i} + \left( 7 x + 6 y \right) \mathbf{j}
f \left( x, y \right) =

B. \mathbf{F} \left( x, y \right) = -5 y \mathbf{i} - 4 x \mathbf{j}
f \left( x, y \right) =

C. \mathbf{F} \left( x, y, z \right) = -5 x \mathbf{i} - 4 y \mathbf{j} + \mathbf{k}
f \left( x, y, z \right) =

D. \mathbf{F} \left( x, y \right) = \left( -5 \sin y \right) \mathbf{i} + \left( 14 y - 5 x \cos y \right) \mathbf{j}
f \left( x, y \right) =

E. \mathbf{F} \left( x, y, z \right) = -5 x^{2} \mathbf{i} + 7 y^{2} \mathbf{j} + 3 z^{2} \mathbf{k}
f \left( x, y, z \right) =

Note: Your answers should be either expressions of x, y and z (e.g. "3xy + 2yz"), or the letter "N"

Answer 1

(a)

Conservative

(b)

Not conservative

(c)

Conservative.

Explanation:

(a)

`\mathbf{F}(x,y) = (-10x+7y,7x+6y)`

Notice that

`\frac{\partial\mathbf{F}_y}{\partial x} = 7`

and

`\frac{\partial\mathbf{F}_x}{\partial y} = 7`

Therefore the field is conservative.

(b)

Notice that

`\mathbf{F}(x,y) = (-5y,-4x)`

and

`\frac{\partial\mathbf{F}_y}{\partial x} = -4`

but

`\frac{\partial\mathbf{F}_x}{\partial y} = -5`

Therefore is not conservative.

(c)

Notice that

To prove that the vector field is conservative you have to compute the curl of the vector field and you would get that.

`\mathbf{F}(x,y,z) = (-5x,-4y,1)`

`\\abla * \mathbf{F} = (0,0,0)`

Therefore your vector field is conservative.