Integral Curves of Vector Field B in $\mathbb{R}^3$

[WannabeNewton]

Homework Statement

For $$\mathbb{R}^{3}$$ find the integral curves of the vector field $$B = xy\frac{\partial }{\partial x} - y^{2}\frac{\partial }{\partial y}$$.

Homework Equations

The Attempt at a Solution

I am having a hard time understand just how to set up the differential equations in order to solve for the family of integral curves. Is it:
$$\frac{\mathrm{d} x}{\mathrm{d} t} = xy$$
$$\frac{\mathrm{d} y}{\mathrm{d} t} = -y^{2}$$
$$\frac{\mathrm{d} z}{\mathrm{d} t} = 0$$?

 

[Mark44]

Homework Statement

For $$\mathbb{R}^{3}$$ find the integral curves of the vector field $$B = xy\frac{\partial }{\partial x} - y^{2}\frac{\partial }{\partial y}$$.

I'm pretty rusty on vector calculus, but this doesn't look like a vector field to me. Are you sure this is the exact wording of the problem?

Homework Equations

The Attempt at a Solution

I am having a hard time understand just how to set up the differential equations in order to solve for the family of integral curves. Is it:
$$\frac{\mathrm{d} x}{\mathrm{d} t} = xy$$
$$\frac{\mathrm{d} y}{\mathrm{d} t} = -y^{2}$$
$$\frac{\mathrm{d} z}{\mathrm{d} t} = 0$$?

 

[WannabeNewton]

Yes sir it was. Its the only problem in Appendix B (Diffeomorphisms and Lie Derivatives) of Spacetime and Geometry - S. Carroll.