Stoke's Theorem: Can't Solve (b)? Get Answers Now!

• chickens
In summary, the problem with part (b) is that the triangle is not broken into three sides. The integrator is not for 0\le x\le 1, 0\le y\le 1; you want to integrate over the triangle with boundaries y= 0, x= 1, y= x.
chickens
Hi guys... I'm stuck in this question. Its not a homework nor coursework, just practice. The answer for both is the same, that is -1 ...I'm able to get (a) but (b) ... still wondering what went wrong. Please enlighten me. Thanks in advance.

Last edited:
That ds=dxdy/2 part.. it's not how it works. what if I wanted to integrate over a circle? would you take ds=pi (dx²+dy²)?

chickens said:
Hi guys... I'm stuck in this question. Its not a homework nor coursework, just practice. The answer for both is the same, that is -1 ...I'm able to get (a) but (b) ... still wondering what went wrong. Please enlighten me. Thanks in advance.
View attachment 8120

Also, how can you say this:
ds = 1/2dxdy

You are saying that ds is a vector. Yet, 1/2dxdy is not.

sorry... that ds suppose to have z_hat with it

quasar987 said:
That ds=dxdy/2 part.. it's not how it works. what if I wanted to integrate over a circle? would you take ds=pi (dx²+dy²)?

i'm not sure about the ds part as usually there is this table i would refer to, and it says ds = dxdy z_hat but ... I've also tried doing using it, i won't get the same answer.

If need to integrate over a circle, transform the vector to cylindrical type and use cylindrical variables r phi z to get it?

In tutorial classes, they gave a semicircle and that was quite easy to get it... but this geometrical shape triangle... is my first time trying it, and i got so say... its cracking my head (i know its simple stuff...but i really don't get where went wrong)

First, at least one of the arrows in your picture is going the wrong way. I suspect that your contour is supposed to go FROM (1, 1) T0 (0, 0), not the otherway round as you have it.

$d\vec{S}$ is NOT "$\frac{1}{2} dxdy \vec{z}$". The fact that you are integrating over a triangle has nothing to do with the differential of area (everything to do with limits of integration!). Since this is in the xy-plane, $d\vec{S}= dxdy \vec{z}$. The integral is NOT for $0\le x\le 1$, $0\le y\le 1$; that would be over the unit square. You want to integrate over the triangle with boundaries y= 0, x= 1, y= x.

For part (b) break the triangle into the three sides. Write parametric equations for each.

HallsofIvy said:
First, at least one of the arrows in your picture is going the wrong way. I suspect that your contour is supposed to go FROM (1, 1) T0 (0, 0), not the otherway round as you have it.

$$d\vec{S}$$ is NOT "$$\frac{1}{2} dxdy \vec{z}$$". The fact that you are integrating over a triangle has nothing to do with the differential of area (everything to do with limits of integration!). Since this is in the xy-plane, $$d\vec{S}= dxdy \vec{z}$$. The integral is NOT for $$0\le x\le 1$$, $$0\le y\le 1$$; that would be over the unit square. You want to integrate over the triangle with boundaries y= 0, x= 1, y= x.

For part (b) break the triangle into the three sides. Write parametric equations for each.

yup...sorry... the contour suppose to go from (1,1) to (0,0) thanks for the mistake pointed out :)

anyway...mind reedit the latex thing... can't seem to get them =_=

1. What is Stoke's Theorem?

Stoke's Theorem is a fundamental theorem in vector calculus that relates the surface integral of a vector field over a surface to the line integral of the same vector field around the boundary of the surface.

2. How is Stoke's Theorem used?

Stoke's Theorem is used to evaluate surface integrals by converting them to line integrals, which are often easier to compute. It is also used to prove other theorems in vector calculus, such as the Divergence Theorem.

3. What is the formula for Stoke's Theorem?

The formula for Stoke's Theorem is ∫∫S curl(F) · dS = ∫C F · dr, where S is the surface, C is the boundary of the surface, F is the vector field, curl(F) is the curl of the vector field, dS is the differential of the surface element, and dr is the differential of the curve element.

4. Can you give an example of applying Stoke's Theorem?

One example of applying Stoke's Theorem is to find the work done by a force field on an object moving along a closed path. By using Stoke's Theorem, the line integral of the force field can be converted to a surface integral, making it easier to calculate.

5. What is the difference between Stoke's Theorem and Green's Theorem?

Stoke's Theorem is a generalization of Green's Theorem, which is used for evaluating line integrals over a closed curve in the xy-plane. Stoke's Theorem extends this concept to evaluate surface integrals over any surface, not just those in the xy-plane. Additionally, Green's Theorem only applies to 2D vector fields, while Stoke's Theorem applies to 3D vector fields.

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