Does This Line Integral Depend Only on Its End Points?

[bugatti79]

Homework Statement

Show that the following line integral depends only on the end points of C by using a suitable theorem.

Then evaluate without parameterising it.

Homework Equations

\int_{(0,0)}^{(3,2)} 2xe^y dx + x^2e^y dy

The Attempt at a Solution

Is this the theorem...

\displaystyle \int F(x,y) dr =\int_C \frac{\partial \phi}{\partial x} dx +\frac{\partial \phi}{\partial y} dy =\int_a^b (\frac{\partial \phi}{\partial x} \frac{dx}{dt} +\frac{\partial \phi}{\partial y} \frac{dy}{dt}) dt...?

 

[lanedance]

do you know about conservative functions and vector fields ?

 

[bugatti79]

do you know about conservative functions and vector fields ?

Yes, this force vector field is conservative because g_x=f_y, hence is independent of path C..but I don't know what theorem to use?

 

[lanedance]

how about the gradient theorem

 

[bugatti79]

how about the gradient theorem

It must be simpler than I thought

I calculate the potential function \phi (x,y)= x^2 e^y +C. we know the gradient of the potential function is the force vector field given in the question. I integrated backwards to get the potential, s I guess that's the theorem..right?

I evaluate the force field to be 9e^2...based on \displaystyle \int \nabla \phi \dot dr = \phi(x_1, y_1) - \phi (x_0, y_0)...

 

[lanedance]

looks good