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Determine if Vector Field is Conservative and Find the Potential of a Conservative Vector Field Example 3
Often you may be asked in Calculus 3 (or sometimes called “vector” calculus) to determine whether a vector field is conservative or not, and if it is, to find the potential of it.
Example problem: Determine if F is a conservative vector field and if it is, find the potential of the vector field, given \(F(x,y) = {e^x}\sin (y)\mathop i\limits^ \to + {e^x}\cos (y)\mathop j\limits^ \to \)
Let the i component be represented by P, and the j component represented by Q.
Just find the partial derivatives of each component.
In this example, i needs to be derived with respect to y, and j derived with respect to x.
\[\begin{array}{l}\frac{{\partial Q}}{{\partial x}} = {e^x}\cos (y)\\\frac{{\partial P}}{{\partial y}} = {e^x}\cos (y)\\\frac{{\partial Q}}{{\partial x}} = \frac{{\partial P}}{{\partial y}}\end{array}\]
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.
\[f = \int {({e^x}\sin (y)} )dx\]
\[f = {e^x}\sin (y) + g(y)\]
Where g(y) is any function of y.
Now derive this result with respect to y.
\[\frac{{\partial f}}{{\partial y}} = {e^x}c\cos (y) + g'(y)\]
This should look like the j component in F.
It does, with the exception of g'(y).
So, g'(y) = 0.
Integrating that, g(y) = K, where K is a real-number constant.
So the potential f is,
\[f(x,y) = {e^x}\sin (y) + K\]
And you can check it yourself, find the partial derivative with respect to x and you’ll get the i component in F.
If find the partial derivative with respect to y and you’ll get the j component in F.
Click here to try another type of example, work by a vector field onto a particle