Finding Potential of Conservative Vector Field Ex2

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Determine if Vector Field is Conservative and Find the Potential of a Conservative Vector Field Example 2

We can try another problem regarding vector fields, you sometimes may be asked to determine whether the vector field is conservative or not, and if it is, you’re then asked to find the potential.

You’re almost certainly going to find out that the vector field is conservative, BUT, if you’re given maybe 5 of these kinds of questions on an examination, then there’s definitely a chance one of them might not be conservative. You do need to have a vector field that’s conservative to find the potential.

Example problem: Determine if F is a conservative vector field and if it is, find the potential, given:

\[\mathop F\limits^ \to = \left( {\ln (y) + 2x{y^3}} \right)\mathop i\limits^ \to + \left( {3{x^2}{y^2} + \frac{x}{y}} \right)\mathop j\limits^ \to \]

Let the i component be represented by P, and the j component represented by Q.

Just find the partial derivatives of each component, i needs to be derived with respect to y, and j derived with respect to x.

\[\begin{array}{l}\frac{{\partial Q}}{{\partial x}} = 6x{y^2} + \frac{1}{y}\\\frac{{\partial P}}{{\partial y}} = \frac{1}{y} + 6x{y^2}\\\frac{{\partial Q}}{{\partial x}} = \frac{{\partial P}}{{\partial y}}\end{array}\]

Since these partial derivatives are equal to each other, the vector field F provided is indeed a conservative vector field.

To find the potential f, take the i component and integrate with respect to x, then derive that with respect to y.

\[f = \int {(ln (y) + 2x{y^3}} )dx\]

\[f = xln (y) + {x^2}{y^3} + g(y)\]

Where g(y) is any function of y.

Now derive this result with respect to y.

\[\frac{{\partial f}}{{\partial y}} = \frac{x}{y} + 3{x^2}{y^2} + g'(y)\]

This should look like the j component in F.

It does, with the exception of g'(y).

So, g'(y) = 0.

But integrating that, g(y) = K, where K is a real-number constant.

So the potential f is,

\[f(x,y) = xln (y) + {x^2}{y^3} + K\]

And you can check it yourself, find the partial derivative with respect to x and you’ll get the i component in F.

If you find the partial derivative with respect to y and you’ll get the j component in F.

Click here to try another example where you find potential and work of a conservative vector field

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