Question

Determine whether or not the vector field is conservative.
F(x, y)= <-ye⁻ˣʸ, -xe⁻ˣʸ>

Answer 1

The vector field F(x, y)= <-ye⁻ˣʸ, -xe⁻ˣʸ> is conservative because the partial derivatives of its components with respect to x and y are equal.

Step-by-step explanation:

To determine whether the vector field F(x, y)= <-ye⁻ˣʸ, -xe⁻ˣʸ> is conservative, we can apply the mathematical condition that in two dimensions, for a force F to be conservative, the partial derivative of the x-component of the force with respect to y must equal the partial derivative of the y-component of the force with respect to x.

Let's calculate the partial derivatives:

• The partial derivative of the x-component Fx with respect to y is d(-ye⁻ˣʸ)/dy, which gives us -e⁻ˣʸ - xy e⁻ˣʸ.
• The partial derivative of the y-component Fy with respect to x is d(-xe⁻ˣʸ)/dx, which gives us -e⁻ˣʸ - xy e⁻ˣʸ.

Since these partial derivatives are equal, the vector field is indeed a conservative force.