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Determine whether or not F(x,y) = 4e^{-4x} \sin(-4y) i - 4e^{-4x} \cos(-4y) j is a...

Question:

Determine whether or not {eq}\mathbf F(x,y) = 4e^{-4x} \sin(-4y) \mathbf i - 4e^{-4x} \cos(-4y) \mathbf j {/eq} is a conservative vector field.

Conservative Vector Field:

If {eq}\vec F = P\,\vec i + Q\,\vec j {/eq} be a vector field.If P and Q have continuous first order partial derivatives and {eq}\displaystyle \frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}} {/eq} then the vector field F is conservative.

Answer and Explanation:

{eq}\displaystyle \mathbf F(x,y) = 4e^{-4x} \sin(-4y) \mathbf i - 4e^{-4x} \cos(-4y) \mathbf j\\ P = 4e^{-4x} \sin(-4y)\\ Q = - 4e^{-4x} \cos(-4y)\\ \displaystyle \frac{{\partial P}}{{\partial y}} = 4e^{-4x} \frac{{\partial \sin(-4y)}}{{\partial y}}\\ \displaystyle \frac{{\partial P}}{{\partial y}} = -16e^{-4x} \cos(-4y)\\ \displaystyle \frac{{\partial Q}}{{\partial x}} = - 4 \cos(-4y) \frac{{\partial e^{-4x} }}{{\partial x}}\\ \displaystyle \frac{{\partial Q}}{{\partial x}} = - 4 \cos(-4y) (-4) e^{-4x}\\ \displaystyle \frac{{\partial Q}}{{\partial x}} = 16 \cos(-4y) e^{-4x}\\ \displaystyle \frac{{\partial P}}{{\partial y}} \neq \displaystyle \frac{{\partial Q}}{{\partial x}} {/eq}

Thus given vector field is not conservative.


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Solving Partial Derivative Equations

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Chapter 14 / Lesson 1
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