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Vector calculas

  • #1

Homework Statement


Evaluate [tex]\int[/tex] 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.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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Hi mamma_mia66! :smile:
Evaluate [tex]\int[/tex] 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) :wink:
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
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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 [itex](x_1,y_1)[/itex] and [itex](x_2,y_2)[/itex] is just [itex]f(x_2,y_2)- f(x_1,y_1)[/itex]

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

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