Solution:
The vector field is conservative and its potential function is

Step-by-step explanation:
Given the vector field
. Determine whether or not it is conservative and if so find the potential function.
is said to be conservative if the
.
To find the
...
![curl \vec F=\left[\begin{array}{ccc}\hat i&\hat j&\hat k\\dx&dy&dz\\P&Q&R\end{array}\right]\\\\ P=y \\\\ Q=x \\\\ R=z^2](https://img.qammunity.org/2024/formulas/physics/high-school/nyc3ord1f21sncsi8czfvd8ul2ija8zuy0.png)
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![curl \vec F=\left[\begin{array}{ccc}\hat i&\hat j&\hat k\\dx&dy&dz\\y&x&z^2\end{array}\right]\\\\=[(z^2)(dy)-(x)(dz)]\hat i-[(z^2)(dx)-(y)(dz)]\hat j+[(x)(dx)-(y)(dy)]\hat k\\\\=[0-0]\hat i-[0-0]\hat j+[1-1]\hat k\\\\=0\hat i-0\hat j+0\hat k \Longrightarrow \boxed{curl\vec F= < 0,0,0 > \therefore Conservative}](https://img.qammunity.org/2024/formulas/physics/high-school/9dvh4opsorboujz04u2k14k1o1pv257bbq.png)
The vector field is conservative. Now we must find its potential function. Follow these steps to find the potential function.
Integrating the P term with respect to x:

Integrating the Q term with respect to y:

Integrating the R term with respect to z:

The potential function is made up of the results of the above integrals ignoring any repeat terms.

Thus, the problem is solved.