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Determine whether the vector field is conservative. Justify your answer. { F(x,y) = 5y^2 (yi +...

Question:

Determine whether the vector field is conservative. Justify your answer.

{eq}F(x,y) = 5y^2 (yi + 3xj) {/eq}

Conservative Vector Field:

A vector filed {eq}\vec F = P\,\vec i + Q\,\vec j {/eq} is conservative if P and Q have continuous first order partial derivatives and {eq}\displaystyle \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}} {/eq}

Answer and Explanation:

{eq}F(x,y) = 5y^2 (yi + 3xj)\\ F(x,y) =5y^3 i + 15y^2xj\\ \displaystyle \frac{{\partial P}}{{\partial y}}= \frac{{\partial 5y^3}}{{\partial y}}\\ \displaystyle \frac{{\partial P}}{{\partial y}}= 15y^2\\ \displaystyle \frac{{\partial Q}}{{\partial x}}=\frac{{\partial 15y^2x}}{{\partial x}}\\ \displaystyle \frac{{\partial Q}}{{\partial x}}=15y^2\\ \displaystyle \frac{{\partial Q}}{{\partial x}}= \frac{{\partial P}}{{\partial y}} {/eq}

Therefore given vector filed is conservative.


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Solving Partial Derivative Equations

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