MHB Multivariable calculus line integral work

[kenporock]

calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by
gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)

 

[I like Serena]

calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by
gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)

Hi kenporock,

The work is:
$$\int_\gamma \mathbf F(\mathbf s)\cdot d\mathbf s = \int_\gamma \mathbf F(\boldsymbol\gamma(t)) \cdot d\boldsymbol\gamma(t) =
\int_0^1 \mathbf F(\boldsymbol\gamma(t))\cdot \boldsymbol\gamma'(t)\, dt$$

But before we go there, weren't you studying conservative force fields?
Is this one?
That is, can we find a potential energy function $V(x,y)$ such that $\mathbf F(x,y)=\pd V x \mathbf i + \pd V y \mathbf j$?

 

[kenporock]

I am extremely grateful to you.