## Luigi0210 one year ago Stoke's Theorem:

1. Luigi0210

Use Stoke's theorem to solve $$\large \int_{C} F~ dr$$ where F(x,y,z) =<xz, 2xy, 3xy> and C is the boundary of part of plane 3x+y+z=3 in the first octant.

2. Astrophysics

I had trouble understanding Stoke's and I've always wanted to go over it again and learn it a better way...mhm lets see if we can figure this out $\int\limits_{C} \vec F \cdot d \vec r = \int\limits \int\limits_S curl \vec F \cdot d \vec S$ right

3. Luigi0210

Not sure, I'm a bit confused because of these: http://prntscr.com/7t4ipd

4. Luigi0210

Same thing?

5. dan815

yes all same thing

6. Astrophysics

Yes, you should understand the notation and what it means then I think it would be more clear hmm. I'm thinking we can pick any surface S with the boundary C.

7. Astrophysics

This would be a nice question to observe haha, I sort of have an idea, but I think @dan815 @ganeshie8 @Empty would be better in teaching this, plus I can learn from them to xD.

8. dan815

|dw:1436995421444:dw|

9. Astrophysics

Projection?!

10. dan815

we gotta see those intersections with xz yz and xy plane

11. dan815

we can see the 3 intersections with the 3 axis by setting x,y=0 y,z=0 x,z=0

12. dan815

|dw:1436996249413:dw|

13. dan815

thre u go not integrate over that surface area

14. dan815

|dw:1436996425783:dw|

15. dan815

F is a function of x,y so we want to refine our new surface area in terms of x and y too

16. dan815

ds=|F_x X Fy| dxdy

17. dan815

i dont remember that exactly, u can work it out though

18. dan815

|dw:1436996594088:dw|

19. dan815

there fore ds= |fx X fy| dxdy

20. dan815

your domain will be that triangle

21. dan815

|dw:1436996772371:dw|

22. dan815

|dw:1436996788233:dw|

23. dan815

|dw:1436996930190:dw|

24. dan815

finally note how Gx and Gy vectors are formed

25. dan815

|dw:1436997056399:dw|

26. dan815

|dw:1436997128852:dw|

27. dan815

and finally u can look up a proof for why the determinant of the 2 vectors gives you the area composed by them

28. dan815

but for starters u can take the formula at face cvalue how u dot v = |U||V| cos theta and U X V = |U||V| sin theta <--- area

29. dan815

as |V| sin theta is the height

30. dan815

so that would the formula for the area of a parallelogram