Evaluation of a Conservative Vector Fields

[Unart]

Homework Statement

Evaluate closed integral ∫sin(x)dx+zcos(y)dy+sin(y)dz where c is an ellipse $$4x^2+9y^2=36$$, oriented clockwise.

Homework Equations

∫Fds=∫F(c(t))*||c'(t)||
or ∫Fdot ds= ∫F(c(t))*c'(t) where it's clockwise...-∫F(c(t))*c'(t)

The Attempt at a Solution

I don't know where to go with this problem... or whether these equations are relevant with with closed vectors fields
Assuming that I am supposed to find c(t)=<9cos(t),4sin(t)>

But again, I could be mistaken... any help on this topic would be greatly, very greatly appreciated.

 

[mighty2000]

Hello, as a scientist, I would like to offer some guidance on how to approach this problem. First, let's clarify the given information. The closed integral is ∫sin(x)dx+zcos(y)dy+sin(y)dz and the curve c is an ellipse with equation 4x^2+9y^2=36, oriented clockwise.

To evaluate this integral, we can use the Green's theorem, which states that for a closed curve c, the closed integral of a vector field F can be expressed as the double integral of the curl of F over the region enclosed by c. In other words, ∫Fdot ds= ∬(∂Fz/∂y - ∂Fy/∂z)dA, where dA is the area element.

Now, let's break down the given integral into its component parts: ∫sin(x)dx, zcos(y)dy, and sin(y)dz. We can rewrite these as ∫Fxdx, ∫Fydy, and ∫Fzdz, respectively, where Fx=sin(x), Fy=zcos(y), and Fz=sin(y).

Next, we need to find the curl of each component of F. Using the identities for partial derivatives, we get ∂Fx/∂z=0, ∂Fy/∂x=0, and ∂Fz/∂y=cos(y).

Now, we can plug these values into the Green's theorem and integrate over the region enclosed by c, which is the ellipse 4x^2+9y^2=36. We can use the parametrization c(t)=<9cos(t),4sin(t)> to express the double integral in terms of t.

Finally, we need to consider the orientation of the curve c. Since it is oriented clockwise, we need to change the sign of the integral. Therefore, the final solution is -∫F(c(t))*c'(t)dt, where c(t)=<9cos(t),4sin(t)> and t goes from 0 to 2π.

I hope this helps guide you in the right direction. Remember to always check your work and double check your equations to ensure accuracy. Good luck!