Please help URDetermine whether or not F is a conservative vector field. If it is, find a...

Question:

Determine whether or not {eq}\vec{F}(x,y)=\left \langle ye^{x} + \sin y, e^x + x \cos y \right \rangle {/eq} is a conservative vector field. If it is, find a function {eq}f {/eq} such that {eq}\vec{F} = \nabla f {/eq}.

Conservative and Irrotational Vector Fields:

A vector field {eq}\vec{F}=\left<u(x,y),v(x,y)\right> {/eq} is said to be conservative if {eq}\vec{F}=\nabla f {/eq} for some function {eq}f {/eq}: that is, if there is some function {eq}f {/eq} such that {eq}f_x=u {/eq} and {eq}f_y=v {/eq}. Such a vector field is said to be irrotational if {eq}u_y=v_x {/eq}. If {eq}\vec{F} {/eq} is conservative and its partial derivatives are continuous, then {eq}\vec{F} {/eq} is irrotational. Similarly, if {eq}\vec{F} {/eq} is irrotational, its partial derivatives are continuous, and {eq}\vec{F} {/eq} is defined on all of {eq}\mathbb{R}^2 {/eq}, then {eq}\vec{F} {/eq} is conservative.

The vector field {eq}\vec{F}(x,y)=\left< ye^{x} + \sin y, e^x + x \cos y \right> {/eq} is defined on all of {eq}\mathbb{R}^2 {/eq}. As it is...

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