Calculating Integral of C F dot dr Using Stoke's Theorem

In summary, the conversation discusses the process of calculating the integral of a vector field, F, along a given triangle, C, using Stokes' Theorem. The curl of F is first calculated and then used to find the dot product with the normal vector, N, resulting in the integral of 3+6y. The final answer is determined to be 14.
  • #1
Dustinsfl
2,281
5

Homework Statement


Let C be the triangle from (0,0,0) to (2,0,0) to (0,2,1) to (0,0,0) which lies in th plane z=y/2. If F(x,y,z)=-3y^2i+4zj+6xk, calculate integral of C F dot dr using Stoke's Theorm.


Homework Equations


For the curl of F, I obtained -4i-6j+6yk.
For N, I obtained 0i-1/2j+k.
0 greater than equal to X greater than equal to 2.
0 greater than equal to Y greater than equal to 2-x.


The Attempt at a Solution


For F dot N, I obtained 3-6y.
Integrated dydx.
After integration, I obtain -2; however, I believe the answer is 14.
What is going wrong?
 
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  • #2
Hi Dustinsfl! :smile:
Dustinsfl said:
For the curl of F, I obtained -4i-6j+6yk.

No, you're only using half the curl formula. :wink:
 
  • #3
What is the other half then?
 
  • #4
eg ∂Fz/∂y - ∂Fy/∂z
 
  • #5
Here is what I obtained then (0-4)i-(6-0)j+(0--6y)k which simplified to -4i-6j+6yk
 
  • #6
oops!

Sorry, I'm being daft. :redface:

Your curl is correct.

But shouldn't this be 3+6y ? …
Dustinsfl said:
For F dot N, I obtained 3-6y
 
  • #7
Yes that was the problem. I have 14 now, thanks.
 

Related to Calculating Integral of C F dot dr Using Stoke's Theorem

1. What is the formula for calculating the integral of C F dot dr using Stoke's Theorem?

The formula is ∫C F · dr = ∫S (∇ x F) · dS, where C is a closed curve, F is a vector field, and S is any surface bounded by C.

2. What is the significance of using Stoke's Theorem to calculate the integral?

Stoke's Theorem allows us to relate a line integral (integral of a vector field along a closed curve) to a surface integral (integral of the curl of the vector field over a surface bounded by the curve). This can make certain calculations easier and can also provide a deeper understanding of the relationship between the vector field and the curve.

3. How is the direction of the curve C related to the direction of the surface S in Stoke's Theorem?

The direction of the curve C is related to the direction of the surface S by the right-hand rule. If you curl your right hand fingers in the direction of the curve, your thumb points in the direction of the surface normal for S.

4. Can Stoke's Theorem be applied to any closed curve and surface?

Yes, Stoke's Theorem can be applied to any closed curve and surface as long as the curve is smooth and the vector field is continuous in the region bounded by the curve and surface.

5. How does Stoke's Theorem relate to other theorems in multivariable calculus, such as Green's Theorem and the Divergence Theorem?

Stoke's Theorem is a higher-dimensional extension of Green's Theorem in the plane, and is also related to the Divergence Theorem in three dimensions. These theorems all relate a line integral to a surface integral, but Stoke's Theorem is the most general, as it can be applied to any closed curve and surface in any number of dimensions.

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