To determine if the given vector field $\mathbf{F}(x, y) = (4x^3y^3 + 3)\mathbf{i} + (3x^4y^2 + 1)\mathbf{j}$ is conservative and, if so, find a potential function $\phi(x, y)$, we follow these steps:

Step 1: Check if $\mathbf{F}(x, y)$ is conservative.

A vector field $\mathbf{F}(x, y) = P(x, y)\mathbf{i} + Q(x, y)\mathbf{j}$ is conservative if:

$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x},$ where $P(x, y) = 4x^3y^3 + 3$ and $Q(x, y) = 3x^4y^2 + 1$.

Compute $\frac{\partial P}{\partial y}$:

$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(4x^3y^3 + 3) = 12x^3y^2.$

Compute $\frac{\partial Q}{\partial x}$:

$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(3x^4y^2 + 1) = 12x^3y^2.$

Since $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, the vector field $\mathbf{F}(x, y)$ is conservative.

Step 2: Find the potential function $\phi(x, y)$.

For a conservative field, $\mathbf{F} = \nabla \phi$, so:

$\frac{\partial \phi}{\partial x} = P(x, y) = 4x^3y^3 + 3,$ $\frac{\partial \phi}{\partial y} = Q(x, y) = 3x^4y^2 + 1.$

Integrate $\frac{\partial \phi}{\partial x}$ with respect to $x$:

$\phi(x, y) = \int (4x^3y^3 + 3) \, dx = x^4y^3 + 3x + C(y),$ where $C(y)$ is an arbitrary function of $y$.

Differentiate $\phi(x, y)$ with respect to $y$ and compare with $Q(x, y)$:

$\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y} \left( x^4y^3 + 3x + C(y) \right) = 3x^4y^2 + C'(y).$

Equating this to $Q(x, y) = 3x^4y^2 + 1$, we find: $C'(y) = 1 \implies C(y) = y + C_0,$ where $C_0$ is a constant.

Thus, the potential function is: $\phi(x, y) = x^4y^3 + 3x + y + C_0.$

Final Answer:

• The vector field is conservative.
• The potential function is: $\phi(x, y) = x^4y^3 + 3x + y + C_0,$ where $C_0$ is an arbitrary constant.

Do you want further details or clarifications? Let me know!

5 Related Questions:

1. What are the conditions for a vector field to be conservative in three dimensions?
2. How can you determine if a vector field is conservative using path independence?
3. What are practical applications of potential functions in physics or engineering?
4. How does the curl of a vector field relate to it being conservative?
5. Can a vector field be conservative in a domain that is not simply connected?

Tip:

Always verify if the region of interest for the vector field is simply connected—this ensures that being curl-free implies conservativeness.