Can we apply the component test for the given field?

Question

Consider the following vector field $\mathbf F$, where $$\mathbf F=x^3\sin\left(\frac1x\right)\mathbf{i}+(2x+e^y)\mathbf{j}+\left(\frac{x^2-y^2}{x^2+y^2}\right)\mathbf{k}$$ defined on its domain. Can you apply the Component Test for Conservative Fields to check whether the vector field is conservative or not. If yes, then use it to check whether $\mathbf F$ is conservative or not. If not, explain.

I understand that to apply the component test for any field, the domain must be simply connected. But for such a field, how can we determine whether the domain is simply connected or not?

Answer 1

Well, first of all, you should say what the domain actually is. I assume it's all the purely imaginary quaternions $x \,\boldsymbol i + y \,\boldsymbol j+ z \,\boldsymbol k$ for which the formula is defined. That would be $Im\mathbb H$ minus the plane spanned by $\boldsymbol j$ and $\boldsymbol k$, which is homeomorphic to $\mathbb R^3$ minus the $(y,z)$-plane, which is homotopy equivalent to the $x$-axis minus the origin by projecting, which has two simply connected components. That should be enough for your component test.

Answer 2

The maximal domain of the given vector field is $ D= \mathbb R^3\setminus \{(x,y,z):x=0\}$. This is not simply connected, since a closed curve around the $x$-axis can not be contracted to a point. However, if you restrict the domain to a simply connected set, e.g., the half space $D_+ = \{(x,y,z):x>0\} $, then you can apply the component test.