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Let F(x,y,z)=yi^+xj^+z2k^ be a vector field. Determine if its conservative, and find a potential if it is.

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Solution:

The vector field is conservative and its potential function is
\bold{f(x,y,z)=xy+(1)/(3)z^3 +C}

Step-by-step explanation:

Given the vector field
\vec F(x,y,z)=y\hat i+x \hat j +z^2 \hat k. Determine whether or not it is conservative and if so find the potential function.


\vec F(x,y,z) is said to be conservative if the
curl\vec F = < 0,0,0 >.

To find the
curl \vec F...


curl \vec F=\left[\begin{array}{ccc}\hat i&amp;\hat j&amp;\hat k\\dx&amp;dy&amp;dz\\P&amp;Q&amp;R\end{array}\right]\\\\ P=y \\\\ Q=x \\\\ R=z^2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


curl \vec F=\left[\begin{array}{ccc}\hat i&amp;\hat j&amp;\hat k\\dx&amp;dy&amp;dz\\y&amp;x&amp;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}

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:


\int\ {P} \, dx \Longrightarrow \int\ {y} \, dx \Longrightarrow \boxed{=xy}

Integrating the Q term with respect to y:


\int\ {Q} \, dy \Longrightarrow \int\ {x} \, dy \Longrightarrow \boxed{=xy}

Integrating the R term with respect to z:


\int\ {R} \, dz \Longrightarrow \int\ {z^2} \, dz \Longrightarrow \boxed{=(1)/(3) z^3}

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


\boxed{f(x,y,z)=xy+(1)/(3)z^3 +C}

Thus, the problem is solved.

User Crozeta
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