Vector Calculus Homework: Evaluate \int F dr

[mamma_mia66]

Homework Statement

Evaluate $$\int$$ F dr
if F(x,y) =(6x² +4y) i + (4x-2y) j and the curve C is a smooth curve from (1,1) to (2,3).

Homework Equations

The Attempt at a Solution

I took partial derivatives with respect of y for the first term and with respect of x for the second term.
4=4 => F is conservative, but from now I am confused how to solvet. I need at least the idea.

I can find the patential function and the plug in the points values (1,1) and (2,3) for y and x .

I am not sure if I can apply the Fundamental Thm of Line Integrals.

I will appreciate any ideas how to finish the problem. Thank you.

 

[tiny-tim]

Hi mamma_mia66!

Evaluate $$\int$$ F dr
if F(x,y) =(6x² +4y) i + (4x-2y) j and the curve C is a smooth curve from (1,1) to (2,3).
…
4=4 => F is conservative, but from now I am confused how to solvet. I need at least the idea.

Since you've proved F is conservative, that means you're free to choose any path …

so just choose whatever looks easiest …

I'd go for either (1,1) to (1,3) to (2,3), or (1,1) to (2,1) to (2,3)

 

[HallsofIvy]

Saying that Fdr is conservative means that there exist a function f(x,y) such that df= F ds and so the integral of Fdr between two points $(x_1,y_1)$ and $(x_2,y_2)$ is just $f(x_2,y_2)- f(x_1,y_1)$

So another way to do this is to find f(x,y) such that
$$df= \frac{\partial f}{dx}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}= F dr$$
evaluated at (2,3) minus the value at (1, 1).
That means you must have
$$\frac{\partial f}{\partial x}= 6x^2+ 4y$$
and
[tex]\frac{\partial f}{\partial y}= 4x- 2y[/itex]
From the first, f(x,y) must equal $2x^2+ 4xy+ g(y)$ since the "constant of integration" must depend on y only. Now, knowing that
$$\frac{\partial f}{\partial y}= \frac{\partial (2x^2+ 4xy+ g(y)}{\partial y}= 4x+ \frac{dg}{dy}= 4x- 2y$$
what must f equal?

 

[mamma_mia66]

Thank you so much . I did finish the problem. I found the potential function and I pluged in the given points for y and x.