Question

The value of $m$ for which the vector field \[ \vec F(x,y)=\big(4x^{m}y^{2}-2xy^{m}\big)\,\mathbf{i}+\big(2x^{4}y-3x^{2}y^{2}\big)\,\mathbf{j} \] is a conservative vector field, is ____________________ (in integer).

Hint:

In 2D, a quick curl test is $\partial M/\partial y=\partial N/\partial x$. When parameters appear, equate powers and coefficients of matching monomials to solve for them.

Answer 1

Correct Answer: 3

Step 1: Use the test for conservativeness in $\mathbb{R^2$.}
For $\vec F=(M,N)$ on a simply connected domain, $\vec F$ is conservative iff \[ \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. \]
Step 2: Compute the mixed partials.
\[ \frac{\partial M}{\partial y} =\frac{\partial}{\partial y}\big(4x^{m}y^{2}-2xy^{m}\big) =8x^{m}y-2m\,x\,y^{m-1}. \] \[ \frac{\partial N}{\partial x} =\frac{\partial}{\partial x}\big(2x^{4}y-3x^{2}y^{2}\big) =8x^{3}y-6xy^{2}. \]
Step 3: Match coefficients and exponents for all $x,y$.
We require \[ 8x^{m}y-2m\,x\,y^{m-1}=8x^{3}y-6xy^{2} \text{for all }x,y. \] Comparing monomials gives \[ m=3 \text{(from }x^{m}y \leftrightarrow x^{3}y\text{),} \] and then \[ -2m=-6\ \Rightarrow\ m=3 \] consistent with the second term ($x\,y^{m-1}\leftrightarrow x\,y^{2}$). Final Answer:\fbox{$3$}